Information Security, Coding Theory and Related Combinatorics: Information Coding and Combinatorics - Volume 29 NATO Science for Peace and Security Series 1607506629, 9781607506621, 9781607506638

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INFORMATION SECURITY, CODING THEORY AND RELATED COMBINATORICS

NATO Science for Peace and Security Series This Series presents the results of scientific meetings supported under the NATO Programme: Science for Peace and Security (SPS). The NATO SPS Programme supports meetings in the following Key Priority areas: (1) Defence Against Terrorism; (2) Countering other Threats to Security and (3) NATO, Partner and Mediterranean Dialogue Country Priorities. The types of meeting supported are generally “Advanced Study Institutes” and “Advanced Research Workshops”. The NATO SPS Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NATO’s “Partner” or “Mediterranean Dialogue” countries. The observations and recommendations made at the meetings, as well as the contents of the volumes in the Series, reflect those of participants and contributors only; they should not necessarily be regarded as reflecting NATO views or policy. Advanced Study Institutes (ASI) are high-level tutorial courses to convey the latest developments in a subject to an advanced-level audience. Advanced Research Workshops (ARW) are expert meetings where an intense but informal exchange of views at the frontiers of a subject aims at identifying directions for future action. Following a transformation of the programme in 2006 the Series has been re-named and reorganised. Recent volumes on topics not related to security, which result from meetings supported under the programme earlier, may be found in the NATO Science Series. The Series is published by IOS Press, Amsterdam, and Springer Science and Business Media, Dordrecht, in conjunction with the NATO Emerging Security Challenges Division. Sub-Series A. B. C. D. E.

Chemistry and Biology Physics and Biophysics Environmental Security Information and Communication Security Human and Societal Dynamics

Springer Science and Business Media Springer Science and Business Media Springer Science and Business Media IOS Press IOS Press

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Sub-Series D: Information and Communication Security – Vol. 29 ISSN 1874-6268 (print) ISSN 1879-8292 (online)

Information Security, Coding Theory and Related Combinatorics Information Coding and Combinatorics

Edited by

Dean Crnkovi University of Rijeka, Rijeka, Croatia and

Vladimir Tonchev Michigan Technological University, Houghton, Michigan, USA

Published in cooperation with NATO Emerging Security Challenges Division

Proceedings of the NATO Advanced Study Institute on Information Security and Related Combinatories Opatija, Croatia 31 May - 11 June 2010

© 2011 The authors and IOS Press. All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior written permission from the publisher. ISBN 978-1-60750-662-1 (print) ISBN 978-1-60750-663-8 (online) Library of Congress Control Number: 2010941318

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LEGAL NOTICE The publisher is not responsible for the use which might be made of the following information. PRINTED IN THE NETHERLANDS

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved.

v

Preface This book contains papers based on lectures presented at the NATO Advanced Study Institute "Information Security and Related Combinatorics", held in the beautiful town of Opatija at the Adriatic Coast of Croatia from May 31 to June 11, 2010. On behalf of all participants, we would like to thank the NATO Science for Peace and Security Programme for providing funds for the conference, as well as the local sponsors, which included the Ministry of Science and Education of the Republic of Croatia, the Croatian Academy of Sciences and Arts, the Primorsko-goranska County, the University of Rijeka and its Mathematics Department, the Foundation of the University of Rijeka, the Society of Mathematicians and Physicists, the Login Co., the Opatija Tourist Board, the City of Opatija, the City of Rijeka, and Brodokomerc.nova. The Advanced Study Institute had fourteen lecturers: K.T. Arasu (USA), C. Colbourn (USA), F. Fuji-Hara (Japan). W. Haemers (The Netherlands), M. Jimbo (Japan), J.D. Key (USA), H. Kharaghani (Canada), C. Lam (Canada), S. Magliveras, (USA), J. Moori (South Africa), T. Shaska (USA), L. Storme (Belgium), V.D. Tonchev (USA), R. Wilson (USA), and was attended by over 60 graduate students and junior scientists from Albania, Armenia, Belgium, Bosnia and Herzegovina, Bulgaria, Croatia, Germany, Italy, Macedonia, The Netherlands, Russia, Turkey, and USA. The unifying theme of the conference was combinatorial mathematics used in applications related to information security, cryptography, and coding theory. The book will be of interest to mathematicians, computer scientists and engineers working in the area of digital communications, as well as to researchers and graduate students who are willing to learn more about the applications of combinatorial mathematics to problems arising in communications and information security. The majority of papers are surveys on topics that are subject to current research and are written in a tutorial text book style that makes this volume a good source as an additional text for a course in discrete mathematics or applied combinatorics. The book can be used in graduate courses of applied combinatorics with a focus on coding theory and cryptography.

Dean Crnkovi´c and Vladimir Tonchev

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Contents Preface Dean Crnkovi and Vladimir Tonchev

v

Crypto Applications of Combinatorial Group Theory Ivana Ili and Spyros S. Magliveras

1

Generating Rooted Trees of m Nodes Uniformly at Random Kenneth Matheis and Spyros S. Magliveras

17

On Jacobsthal Binary Sequences Spyros S. Magliveras, Tran van Trung and Wandi Wei

27

Applications of Finite Geometry in Coding Theory and Cryptography A. Klein and L. Storme

38

The Arithmetic of Genus Two Curves T. Shaska and L. Beshaj

59

Covering Arrays and Hash Families Charles J. Colbourn

99

Sequences and Arrays with Desirable Correlation Properties K.T. Arasu

136

Permutation Decoding for Codes from Designs, Finite Geometries and Graphs J.D. Key

172

Finite Groups, Designs and Codes J. Moori

202

Designs, Strongly Regular Graphs and Codes Constructed from Some Primitive Groups Dean Crnkovi, Vedrana Mikuli Crnkovi and B.G. Rodrigues

231

Matrices for Graphs, Designs and Codes Willem H. Haemers

253

Finding Error-correcting Codes Using Computers Clement Lam

278

Quantum Jump Codes and Related Combinatorial Designs Masakazu Jimbo and Keisuke Shiromoto

285

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Unbiased Hadamard Matrices and Bases Hadi Kharaghani

312

Multi-structured Designs and Their Applications Ryoh Fuji-Hara and Ying Miao

326

Recent Results on Families of Symmetric Designs and Non-embeddable Quasi-residual Designs Mohan S. Shrikhande and Tariq A. Alraqad

363

Codes and Modules Associated with Designs and t-uniform Hypergraphs Richard M. Wilson

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Finite Geometry Designs, Codes, and Hamada’s Conjecture Vladimir D. Tonchev

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Subject Index

449

Author Index

451

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-1

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Crypto applications of combinatorial group theory Ivana Ili´c and Spyros S. Magliveras CCIS, Department of Math. Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA e-mail: [email protected], [email protected]

Abstract. The design of a large number of cryptographic primitives is based on the intractability of the traditional discrete logarithm problem (tDLP). However, the well known quantum algorithm of P. Shor [9] solves the tDLP in polynomial time, thus rendering all cryptographic schemes based on tDLP ineffective, should quantum computers become a practical reality. In [5] M. Sramka et al. generalize the DLP to arbitrary finite groups. The DLP for a non-abelian group is based on a particular representation of a chosen family of groups, and a choice of a class of generators for these groups. In this paper we show that for P SL(2, p) = α, β, p an odd prime, certain choices of generators (α, β) must be avoided to insure that the resulting generalized DLP is indeed intractable. For other types of generating pairs we suggest possible cryptanalytic attacks, reducing the new problem to the earlier case. We note however that the probability of success is asymptotic to p1 as p → ∞. The second part of the paper summarizes our successful attack of the SL(2, 2n ) based Tillich Zémor cryptographic hash function [2], and show how to construct collisions between palindromic strings of length 2n + 2. 2000 Mathematics Subject Classification: 68P25, 94A60. Keywords. Discrete logarithm, finite groups, intractability, representations and presentations of groups, P SL(2, p), public key cryptosystems, Tillich-Zémor hash function.

Introduction In a recent quote, P. Nguyen states “Due to Shor’s algorithms for computing prime factorizations and discrete logarithms on quantum computers, most of present day public key cryptosystems must be considered insecure , if sufficiently large quantum computers became available. ... One interesting line of research in this direction is the use of computational problems in non-abelian groups ... ” [6]. In this article we discuss recent results on the generalized discrete logarithm problem (GDLP) in the family of non-abelian simple groups P SL(2, p), p an odd prime. In particular we examine these groups in their representations as matrices over GF (p), and investigate weak generator choices for the generalized DLP problem. In the second part of the paper we summarize the interest-

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I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

ing approach in [2] which culminated with the demise of the well known Tillich-Zémor cryptographic hash function [13].

1. Preliminaries The authors of [5] generalize the discrete logarithm problem from finite cyclic groups to arbitrary finite groups. We restate the definition. Let G be a finite group generated by α1 , . . . , αt , i.e., G = α1 , . . . , αt . Denote by α = (α1 , . . . , αt ), the ordered tuple of generators of the group G. As defined in [5], for a given β ∈ G, the generalized discrete logarithm problem (GDLP) of β with respect to α is to determine a positive integer k and a (kt)-tuple of non-negative integers x = (x11 , . . . , x1t , . . . , xk1 , . . . , xkt ) such that β=

k 

(α1xi1 . . . αtxit ) .

i=1

We can write this formally as β = αx . The (kt)-tuples (x11 , . . . , x1t , . . . , xk1 , . . . , xkt ) are called the generalized discrete logarithms of β with the respect to α = (α1 , . . . , αt ). Denote by Sk =

k 

i=1

(α1xi1 . . . αtxit ) | xij ∈ Znj



where nj denotes the order of element αj . Then, the smallest positive integer k0 such that for all k ≥ k0 G ⊆ Sk is called the depth of group G with respect to (α1 , . . . , αt ). There could be more than one generalized discrete logarithm of β with respect to α. Actually, there will be infinitely many generalized discrete logarithms: if x is a generalized discrete ′ ′ ′ logarithm of β with respect to α and if αx = 1, then, the catenations x||x and x ||x are also generalized discrete logarithms of β with respect to α. The generalization of the discrete logarithm problem to finite groups has potential applications in cryptography. To be able to construct secure cryptographic primitives based on the generalized discrete logarithm problem in finite groups, care must be taken to ensure that the groups along with their representations and choice of generators have an intractable generalized discrete logarithm problem. The traditional discrete logarithm problem is generally considered computationally intractable. However, there exist groups and their representations in which the problem can be solved efficiently. For example, in Zn , the additive group of integers modulo n, the discrete logarithm can be easily computed. For a given element β in Zn and generator α of Zn , it is easy to find a non-negative integer x such that xα = β. Since α is a generator, gcd(n, α) = 1, and the multiplicative inverse in the ring (Zn , +, ·) of α can be computed by the extended Euclidean algorithm. In general, one may speak of a tractable/intractable GDLP problem for a given infinite family of pairs {(Gℓ , Aℓ )}ℓ∈L indexed by L, where the Gℓ are groups in a common representation ρ, and Aℓ a particular set of generators for Gℓ .

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

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The generalized discrete logarithm problem may be tractable for some groups and generators in representation ρ. We examined the groups P SL(2, p) as potential candidates for cryptographic applications, but our results show that when P SL(2, p) is represented by matrices, the generalized discrete logarithm problem with respect to several types of generating sets does not provide the required strength. As is customary, we denote by Z the ring of integers. We also denote by Z+ the positive integers, and by Z0 the non-negative integers.

2. Generalized discrete logarithm problem in P SL(2, p) Suppose that for an odd prime p the group G = P SL(2, p) is represented by matrices of SL(2, p), up to a factor ±I, where I is the 2 × 2 identity matrix. Suppose further that G is generated by two elements, i.e., G = A, B. We have examined the tractability of the generalized discrete logarithm problem in this setup with respect to different generating pairs of elements (A, B). The results of our research show that the hardness of computation of the generalized discrete logarithm problem will depend not only on the group representation, but also on the choice of generators. To perform a detailed analysis on whether the generalized discrete logarithm can be computed efficiently, we considered the following cases: 1) group G is generated by special elements: A = ( 10 11 ), and B = ( 11 01 ); 2) group G is generated by two elements both of order p; 3) group G is generated by two elements, one of which is of order p; 4) group G is generated by two elements none of  which  is of order p. We have analyzed the first two cases in [4]. Suppose that M = ac db ∈ G, with a, b, c, d ∈ Fp , the field of order p. The matrices

A=

„

11 01

«

,

B=

„

10 11

«

are both of order p, non-commuting and generate G, i.e., G = A, B. Moreover, the authors of [5] show that the depth of group G with the respect to the (A, B) is two, so that the element M ∈ G can be written as M = Ai B j Ak B ℓ . We have Ai B j Ak B  =

„

1 i 01

«„

10 j1

«„

1k 01

«„

10 ℓ1

«

.

Hence, „

ab cd

«

=

„

1 + ij + ℓ ((1 + ij) k + i) (1 + ij) k + i j + ℓ (jk + 1) jk + 1

«

.

By equating corresponding entries in the previous equality we obtain the system of equations

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

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1 + ij + ℓ ((1 + ij) k + i) = a (1 + ij) k + i = b j + ℓ (jk + 1) = c jk + 1 = d

which can be solved for i, j, k, ℓ by computing Gröbner basis of the ideal I =  1 + ℓk + ij + ijkℓ + iℓ − a, k + ijk + i − b, j + jkℓ + ℓ − c, jk + 1 − d  . A Gröbner basis for the above ideal is computed over the set of rational numbers: [ ℓ − jic + ja − c, k + id − b, jibc + ji − jab − a + bc + 1, jid − jb + d − 1, ad − bc − 1 ], which yields the following system of equations: in i, j, k, ℓ ∈ Zp . ℓ − jic + ja − c = 0 k + id − b = 0 jid − jb + d − 1 = 0 whose solutions in i, j, k, l represent the generalized discrete logarithms of M with respect to (A, B). The solutions are given by the following proposition: Proposition 2.1 Let A, B and M be as above. Then, there exists a non-negative integer n < p such that nd − b = 0 over Zp , and such that the 4-tuple (i, j, k, ℓ) with i = n, j = (1 − d)(nd − b)−1 , k = b − nd, ℓ = (1 − d)(nc − a)(nd − b)−1 + c provides a solution to M = Ai B j Ak B ℓ . Proof. It can be directly verified that the given values for i, j, k, ℓ satisfy the above system of equations. The existence of n is ensured since M ∈ P SL(2, p) and hence b and d can not simultaneously be equal to zero. ✷ We have shown that the generalized discrete logarithm problem can be solved efficiently in P SL(2, p) with respect to the special given generators (A, B) as defined above. Further, as in [4], we construct an algorithm for computing the generalized discrete logarithm problem in P SL(2, p) with respect to any two generators of order p. Assume that C, D are two non-commuting elements of order p in P SL(2, p). Then, since any two non-commuting elements of order p from P SL(2, p) generate the whole group, it follows that P SL(2, p) = C, D. To determine non-negative integers i, j, k, ℓ such that: M = C i Dj C k Dℓ , we look for an element g ∈ G which satisfies C = g −1 As g and D = g −1 B t g, for some non-negative integers s, t < p and where A and B are the matrices defined above. Then,

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

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M = C i Dj C k Dℓ = (g −1 As g)i (g −1 B t g)j (g −1 As g)k (g −1 B t g)ℓ = (g −1 Asi g)(g −1 B tj g)(g −1 Ask g)(g −1 B tℓ g) = g −1 Asi B tj Ask B tℓ g Denote by x = si, y = tj, v = sk and w = tℓ. Then, gM g −1 = Ax B y Av B w . Let M1 = gM g −1 . Obviously, M1 ∈ G and M1 = Ax B y Av B w . We have transformed the generalized discrete logarithm problem of P SL(2, p) with respect to (C, D) to the generalized discrete logarithm problem of P SL(2, p) with respect to (A, B) which we are able to solve as described earlier. To determine an element g for which the conditions C = g −1 As g and D = g −1 B t g hold simultaneously, we write the system of equations: gC = As g and gD = B t g, for some non-negative integers s, t < p. Since, g = ( gg13 gg24 ), we obtain a system of equations in g1 , . . . , g4 and s and t from which an element g is determined. The existence of such an element g is ensured since P SL(2, p) acts doubly transitively by conjugation on its (p + 1) Sylow-p subgroups. Then, for any two pairs of p-Sylow subgroups, and hence for the particular pairs (A, B) and (C, D), there exists an element g ∈ G such that (C, D) = (Ag , Bg ) .

The third case in our analysis of hardness of the generalized discrete logarithm problem in P SL(2, p), with respect to a pair of generators, is when one of the generators is of order p. Suppose now that P SL(2, p) = A, B where |A| = p. Note that the order of element B can only be divisor of the order of the group p(p2 − 1)/2. Given an element M ∈ P SL(2, p) our goal is to write M in terms of the generators (A, B). In the construction of a word in A and B that represents element M , we will use the result of the following proposition. Proposition 2.2 If G = P SL(2, p) = A, B where |A| = p, then G = A, AB , where AB = B −1 AB.

Proof. Every two non-commuting elements of order p from P SL(2, p) generate the whole group. So we prove that elements A and AB are non-commuting of order p. Conjugate elements have the same order, so |AB | = |A| = p. Now, suppose that elements A and AB commute. Then, AB is in the centralizer of element A, i.e., AB ∈ CG (A) = A. So, AB = Ai for some i ∈ {0, . . . , p − 1}. But then, B normalizes A, hence, A is a proper normal subgroup of A, B. But P SL(2, p) is simple, thus A, B can not be all of P SL(2, p), a contradiction to the fact that A and B generate G. ✷ The proposition that follows provides an upper bound for the depth of P SL(2, p) with respect to two generators one of which is of order p and its proof provides an algorithm for constructing a word in generators A and B that represents a given element M. Proposition 2.3 Suppose that G = P SL(2, p) = A, B, where |A| = p, with no further assumptions on |B| = m. Then, the depth of G with respect to the generating tuple (A, B) is less than or equal to four.

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Proof. Let C = AB = B −1 AB. By Proposition (2.2) the group P SL(2, p) is generated by elements A and C, both of order p. The generalized discrete logarithm problem can be solved efficiently in P SL(2, p) represented by matrices, with respect to two generators of order p. By the method described earlier, the generalized discrete logarithm (i, j, k, ℓ) can be found such that M = Ai C j Ak C ℓ . To represent the element M in terms of the generators A and B we write the following sequence of equalities. M = Ai C j Ak C ℓ = Ai (B −1 AB)j Ak (B −1 AB)ℓ = Ai B −1 Aj BAk B −1 Aℓ B = Ai B m−1 Aj BAk B m−1 Aℓ B Therefore, the generalized discrete logarithm of M ∈ P SL(2, p) with respect to generating tuple (A, B), where |A| = p and |B| = m is (i, m − 1, j, 1, k, m − 1, ℓ, 1). It follows that every element M from P SL(2, p) = A, B, where |A| = p and |B| = m can be represented as M = Ax1 B y1 Ax2 B y2 Ax3 B y3 Ax4 B y4 for some integers x1 , x2 , x3 , x4 ∈ {0, ..., p − 1} and y1 , y2 , y3 , y4 ∈ {0, ..., m − 1}. The proposition follows. ✷ The described method for writing element M as a word in generators A and B does not assure obtaining the shortest possible word that represents M in these generators. Next, we take a look into a possible strategy for writing an element M of group P SL(2, p) in terms of two generators none of which is of order p. Suppose that we have an efficient method for constructing an element of order p in terms of the generators A and B. In the following proposition we will use the notation wp (A, B) to represent a word in A and B which is of order p as an element of G. Proposition 2.4 If G = P SL(2, p) = A, B where the orders of A and B are relatively prime to p, and if P = wp (A, B), is a word in A and B, of order p as an element of G, then G = A, P  or G = B, P . Proof. Let N be the normalizer in G of P , i.e. N = NG (P ). Then, at least one of the elements A, B is not in N . Otherwise if A, B were both in N , then A, B would be a subgroup of N , that is G = A, B ≤ N , and therefore we would have that N = G. This would imply that P  is a non-trivial, proper, normal subgroup of G, contradicting the fact that G is simple. Without loss of generality, suppose that A ∈ / N. Then A, P  = P SL(2, p), because the only proper subgroups of P SL(2, p) containing P  are subgroups of the normalizer of P . Similarly, if B ∈ / N , it follows that P SL(2, p) = B, P . ✷

If A, B and P are as in Proposition 2.4 we can solve efficiently the generalized discrete logarithm problem with respect to (A, P ) since P SL(2, p) = A, P  and |P | = p. Therefore, we can solve the generalized discrete logarithm problem with respect to (A, B). Given M ∈ P SL(2, p) = A, B and P = wp (A, B) as in the Proposition 2.4 we can write element M as a word in A, B as follows. Without loss of generality, assume that A ∈ / NG . Conjugate element P by element A, i.e., compute P A = A−1 P A. Based

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

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on the Proposition 2.2, P SL(2, p) = P, P A . Based on the proof of the Proposition 2.3, if |A| = s, we have: M = P i (P A )j P k (P A )ℓ = P i As−1 P j AP k As−1 P ℓ A = wp (A, B)i As−1 wp (A, B)j wp (A, B)wp (A, B)k As−1 wp (A, B)ℓ A The direct consequence is that the depth of the P SL(2, p) with respect to the generators both of order relatively prime to p, will depend on the word P = wp (A, B). We examine a bit further possible attacks to the GDLP for G = P SL(2, p) based on Proposition 2.4. A word of shortest possible length in A and B to produce an element of order p is AB or BA. We will consider the case where |A| = |B| = d = (p − 1)/2 and |AB| = p. This condition occurs systematically in P SL(2, p), however, unfortunately for the cryptanalyst, the probability of this occurrence goes to zero as p → ∞.

We will need some well known facts about the group P SL(2, q), q = pm , p an odd prime, which we state below, without proof, as a proposition. In what follows φ stands for Euler’s φ function. Proposition 2.5 Suppose that G = P SL(2, q), q = pm , p an odd prime. Then,

(a) The Sylow-p subgroup of G is elementary abelian of order q, (b) If x ∈ G is of order d, then d divides (q −1)/2, or d = p, or d divides (q +1)/2, (c) There is a single conjugacy class of subgroups of order (q − 1)/2, and these are cyclic. Similarly, there is a single conjugacy class of subgroups of order (q + 1)/2, and they are cyclic. (d) If x ∈ G is of order d = 2 dividing (q ± 1)/2 then x belongs to one and only one cyclic subgroup of G of order (q ± 1)/2. (e) If d = 2 divides (q ± 1)/2 there are φ(q±1) conjugacy classes of element of 2 order d in G. (f) If x ∈ G is of order d|(q ± 1)/2, d = 2, then the centralizer CG (x) is x, while the normalizer NG (x) is dihedral of order q ± 1. We will now examine the very special case where G = P SL(2, p) is generated by two elements of order (p − 1)/2. Similar results can be derived for the other possible cases. In what follows, Let X be the set of all elements of order d = (p − 1)/2 in G. We will consider the action of G by conjugation on X × X. Note that all pairs (A, B) in a G−orbit on X × X share almost all critical properties of interest to our problem, as conjugation by an element g ∈ G induces an automorphism of G. For example if (A, B) generate G so does (A, B)g = (Ag , B g ), for g ∈ G. Similarly, the order of AB is the same as the order of Ag B g = (AB)g , etc. Thus it suffices to examine one representative from each orbit of G on X × X. Since G acts transitively by conjugation on the cyclic subgroups of order (p − 1)/2, without loss of generality, we will select one such subgroup, say C and one fixed generator x ∈ C, so that C = x. Now, CG (x) = {y ∈ G | xy = yx} = x = C. We have the following consequences of Proposition 2.5:

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Proposition 2.6 If G and X are as above, and d = (p − 1)/2, then: (a) |X| = φ(d)p(p + 1)/2, (b) Let x be any fixed element of X. In the action of C = CG (x) on X by conjugation there are exactly φ(d) orbits of length 1, and v = (φ(d)p(p + 1) − 2)/2d orbits of length d. (c) Of the v orbits Oi of length d exactly 2φ(d) − 2 are such that if y ∈ Oi then |xy| = p. Proof. (a) Since each of the φ(d)/2 conjugacy classes of elements of order d has |G|/d = p(p + 1) elements, it follows that |X| = [φ(d)p(p + 1)]/2. (b) C = CG (x) = x has exactly φ(d) elements y of order d in it, and since these elements commute with x, the orbit y C = {y} and has length 1. If y ∈ X \ C then K = CG (y) = y, and K∩C = {1}, hence the orbit y C has exactly |C| elements. Thus, the number of orbits of length d is [(φ(d)p(p+1))/2−φ(d)]/d = [φ(d)(p(p+1)−2]/2d. (c) We will only give an idea about the proof here. The result follows from calculations in the center of the group ring ZG. In particular, if {Ki }ci=1 are the conjugacy classes c of G, they form a basis for the center of ZG and Ki Kj = k=1 aijk Kk , with the aijk computable from the character table of G. We have that X is the sum of the φ(d)/2 classes {Kαi } with elements of order d. Thus, in the group ring, the number of elements in xX of order p is the sum of the coefficients of the two classes Kp− and Kp+ in φ(d)/2 φ(d)/2 1 1 Kαi = p(p+1) Kx Kαi . Since each C−orbit on X \ C are of i=1 i=1 |Kx | Kx length d, we further divide by d for the number of C−orbits. ✷ We are now able to state a proposition which is not of much help to the cryptanalyst, but which lends evidence to the notion that strong generators may be possible for a GDLP based system. Proposition 2.7 Let G = P SL(2, p) and let d, X and x ∈ X be as above. If we select a second element y ∈ X randomly, then the probability that the order of xy is p is 2(φ(d)−1)(p−1) which is of course asymptotic to p2 as p → ∞. φ(d)p(p+1) Proof: Having fixed x ∈ X, by Proposition 2.6 the number of elements y ∈ X such |G| that |xy| = p is 2(φ(d) − 1)d. Since |X| = φ(d) 2 · d we have: P r{|xy| = p} =

2(φ(d) − 1)d

φ(d) 2

· p(p + 1)

=

2(φ(d) − 1)(p − 1) 2 4(φ(d) − 1)d = ❀ φ(d)p(p + 1) φ(d)p(p + 1) p

hence the result. ✷ It is clear of course that if (A, B) ∈ X × X with |AB| = p, then A, B = G.

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3. Relations By solving the generalized discrete logarithm problem for a finite group with respect to a given set of generators we are factorizing group elements in terms of the generators. By equating two different factorizations of the same group element, we obtain a relation. This observation holds in any finite group as we discuss in the next section. Let G be a finite group generated by α1 , . . . , αt , i.e., G = α1 , . . . , αt . Denote by α = (α1 , . . . , αt ) the ordered tuple of generators of the group G. For a given β ∈ G, assume that β=

k 

(α1xi1 . . . αtxit )

i=1

i.e., β = αx , where x = (x11 , . . . , x1t , . . . , xk1 , . . . , xkt ). Recall that x = (x11 , . . . , x1t , . . . , xk1 , . . . , xkt ), the generalized discrete logarithm with respect to the generators α = (α1 , . . . , αt ), is not unique, in fact there will exist s infinitely many distinct y = (y11 , . . . , y1t , . . . , ys1 , . . . , yst ) such that β = αy = i=1 α1yi1 . . . αtyit . For any such y we have: k 

i=1

α1xi1 . . . αtxit =

s 

α1yi1 . . . αtyit .

i=1

In this way we obtain non-trivial relations among the generators. Further, by collecting different relations we may obtain a presentation of the group : G = X|R, where X is the set of generators, and R a set of relations of the above type, sufficiently many to completely determine the group. Relations of particular interest in cryptography are those which represent the identity element of the group, that is of the form 1G = a word in the generators. Moreover, in a finite group G we can always convert a presentation of the form G = X|R, into one of k the form G = X|R′ , where R′ is a set of relations of the type: i=1 α1xi1 . . . αtxit = 1G . k The length of word w = i=1 α1xi1 . . . αtxit in the symbols α1 , . . . , αt , where k t the xij are non-negative integers, is defined to be the integer |w| = i=1 j=1 xij . Moreover, if w1 and w2 are words in the symbols α1 , . . . , αt and ρ : w1 = w2 is a relation, the length of the relation is defined to be the integer |ρ| := |w1 | + |w2 |. If G is a finite group generated by α1 , . . . , αt , a relation ρ in the α1 , . . . , αt is said to be short if |ρ| = O(log (|G|)), otherwise ρ is said to be long. Relations of importance to cryptographic hash functions of the Tillich-Zémor type are those which are short.

We turn to our group of interest, P SL(2, p), and examine the length of some relations there. Let G = P SL(2, p), and consider the elements A = ( 10 11 ), B = ( 11 01 ) in G. The matrices A and B are both of order p, non-commuting and thus generate P SL(2, p). As we have seen earlier, the depth of P SL(2, p) with respect to the generating tuple (A, B)

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10

is two. Therefore, the identity matrix I ∈ P SL(2, p) can be written as I = Ai B j Ak B ℓ for some non-negative integers i, j, k and ℓ. In the next proposition we establish that for any prime p, any relation of the form I = Ai B j Ak B ℓ in P SL(2, p) is long. Proposition 3.1 Let A, B and I be matrices in P SL(2, p) as above. Then, a solution (i, j, k, ℓ) to the generalized discrete logarithm problem I = Ai B j Ak B ℓ is such that either i + j + k + ℓ ≥ p or i = j = k = ℓ = 0. Proof. i

ℓ



1i 01



10 j1



1k 01

j

k



1 + ij + ℓ((1 + ij)k + i) j + ℓ(jk + 1)

AB A B =



10 ℓ1



.

Therefore,

10 01



=

(1 + ij)k + i jk + 1



.

Then, jk + 1 = 1 (mod p) and hence jk = 0 (mod p). By using jk = 0 (mod p), we obtain

1 + ij + ℓk + ℓi k + i 10 = j+ℓ 1 01 So, j + ℓ = 0 (mod p) and k + i = 0 (mod p) i.e., j + ℓ = s1 p, s1 ∈ Z0 and k + i = s2 p, s2 ∈ Z0 . If s1 ≥ 1, then j + ℓ ≥ p. Hence, i + j + k + ℓ ≥ p. If s1 = 0, i.e., j + ℓ = 0, then j = ℓ = 0. Similarly, s2 ≥ 1 leads to i + j + k + ℓ ≥ p, and s2 = 0 leads to k = i = 0. The length of the word 1G = Ai B j Ak B ℓ , is i + j + k + ℓ ≥ p or i = j = k = ℓ = 0. Thus, i + j + k + ℓ ≥ p. ✷

We remark that since for p > 7, p > 3 log p > log(|P SL(2, p)|), any relation of the form I = Ai B j Ak B ℓ is long, for all p > 7.

4. The demise of the Tillich-Zémor hash function Let V = {0, 1}∗ be the Kleene closure of {0, 1}, i.e. the set of all binary sequences of arbitrary but finite length. Moreover, for n ∈ Z+ , denote by Vn = {0, 1}n the set of all binary sequences of length n. For a given parameter n ∈ Z+ , by a hash function we mean any function h : V → Vn . If v ∈ V , we denote by v r the reversal of v, i.e. the reflection of v with respect to a central axis. For example if v = 00111, v r = 11100. Definition 4.1 For a fixed parameter n ∈ Z+ , a hash function h : V → Vn is said to be a cryptographic hash function if h has the following additional properties : 1. preimage resistance: For essentially all y ∈ Vn it is computationally infeasible to find x ∈ V such that h(x) = y,

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2. 2nd-preimage resistance : For any given x ∈ V it is computationally infeasible to determine any x′ ∈ V , such that x′ = x and h(x′ ) = h(x), and 3. collision resistance: It is computationally infeasible to find any x, x′ ∈ V such that x = x′ and h(x) = h(x′ ). It is clear that the three properties are not independent, but if for a given h a cryptanalyst succeeds in breaching any one of the three, then h is considered compromised. However, a satisfactory attack on the collision resistance property must also satisfy a rather severe length requirement that the lengths of x and x′ must be polynomial in the parameter n. In their paper Hashing with SL2 [13], Tillich and Zémor propose a cryptographic hash function based on computing matrix products in the non-abelian group SL(2, q). A brief history of the evolution of the Tillich-Zémor hash function (TZ) is given in the introduction of [2]. We give here a brief description of the scheme in its final form and a summary of the main steps that led to its cryptanalysis. 4.1. The final version of the Tillich-Zémor hash function Input parameters are a positive integer n, and an irreducible polynomial q(x) of degree n over the field of two elements F2 = GF (2). Let F be the finite field of order 2n represented as F = F2 [x]/(q(x)). Let α be a root of q(x) and define s0 :=



α1 10



,

s1 :=



αα+1 1 1



Then the matrices s0 and s1 generate the group G = SL(2, F) of all unimodular matrices ˘ is defined as follows: over F. The Tillich-Zémor hash function h ˘ For bitstring v = b0 b1 · · · bm ∈ V define h(v) := sb0 sb1 · · · sbm ∈ G. ˘ maps a binary string v of arbitrary length to a matrix in G which requires Note that h 4 entries from F, thus maps V to V4n . Any satisfactory attack must work for any n and any irreducible polynomial q(x) of degree n over F2 . Thus, we note that the problem is specific to the representation of F as well as to the generators. The orders k, ℓ of s0 and s1 could be very large, for example any divisors of 2n + 1 or 2n − 1 and can be efficiently calculated. If k or ℓ is small, then the system can be effectively attacked because one can write a short relation, such as sk0 = I, or sℓ1 = I, I the identity of G. Thus a successful attack must assume nothing about the orders k and ℓ. In the proposition that follows we prove the existence of short relations in any finite group G generated by two elements. Proposition 4.1 Let G be a finite group generated by two elements A and B. Then there exist a relation ρ : w1 = w2 where w1 and w2 are two different words in A and B, such that |w1 | + |w2 | ≤ O(log2 |G|).

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Proof. We construct the blocks of all words of successive lengths in A and B. Let B0 = {I}, where I is the identity of the group G. Let Bk be the collection of all words in A and B of length k. Then |Bk | = 2k . n+1 Let n be the positive integer such that and such that k=0 |Bk | > |G| n n n+1 n+1 |B | = 2 − 1 we can write 2 − 1 ≤ |G|, |B | ≤ |G|. Since k k k=0 k=0 n+1 i.e., 2 ≤ |G| + 1. By taking logarithms of both sides of the inequality, we obtain that n + 1 ≤ log2 (|G| + 1). By the pigeon-hole principle, two distinct words, say w1 and w2 belonging to {B0 ∪ B1 ∪ · · · ∪ Bn+1 } must correspond to the same element of G. Then, |w1 | + |w2 | ≤ 2(n + 1) ≤ 2log2 (|G| + 1) = O(log2 (|G|)). ✷

Of course the proof can be generalized to any finite group G generated by k generators. A direct consequence of Proposition 4.1 is that short relations in two generators do exist in SL(2, q). In particular, for G = SL2 (2n ), |G| = 2n (22n − 1), and there are short relations of length at most 6n. The question, of course, is how does one find them ? 4.2. Experimentation

Early cryptanalytic experiments [2] were restricted to cases in which the defining irreducible polynomial q(x) was of degree small enough to allow brute force searching for collisions. Data analysis of experimental results showed that for every input q(x) of degree n, collisions of words of length 2n + 2 were obtained and that among those collisions there were colliding palindromes. Computations were preformed on a standard PC, using computer algebra system Magma [1]. For example, Example 4.1 With irreducible polynomial q(x) = x5 + x4 + x3 + x + 1 used to define the field F25 = F2 [x]/(q(x)) and with Tillich-Zémor generators s0 , s1 , the following collisions of palindromes of length 2n + 2 occur:

palindrome

the same palindrome

    ˘ 00110 01100 0) = h(1 ˘ 00110 01100 1) h(0

 

  v

vr

v

vr

˘ ˘ h(011101101110) = h(111101101111)

Experimental results showed that for each tested choice of F2n = F2 [x]/(q(x)) two bit strings v1 , v2 ∈ {0, 1}n , |v1 | = n, |v2 | = n, with ˘ i v r 0) = h(1v ˘ i v r 1) h(0v i i are obtained.

(i = 1, 2),

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

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4.3. The successful attack It was shown in [2] how to construct collisions between palindromes of length 2n+2 for any defining irreducible polynomial of degree n, that is, pairs (u, v) ∈ V × V such that ˘ ˘ u = v and h(u) = h(v). It was demonstrated that the attack is practical: by constructing collisions for the challenge parameters. The method finds collisions of length a few hundred bits on a standard PC within a second. For the challenge polynomial of largest degree x2039 + x10 + x9 + x8 + x7 + x5 + x4 + x2 + 1 computation still took a few seconds. With very few exceptions we will only state lemmas, propositions or theorems here but will refer to [2] for their proof. 4.3.1. Change of generators Recall that for α a root of irreducible q(x) of degree n in F2 [x] s0 :=



α1 10



,

s1 :=



αα+1 1 1



and ˘ 1 . . . bm ) := sb · · · sb ∈ G h(b m 1 by conjugating the pair of generators (s0 , s1 ) by any element of G we clearly get another pair of generators of G. In particular, conjugating (s0 , s1 ) by s0 yields (ss00 , ss10 ) = (s0 , s−1 0 s1 s0 ) = (c0 , c1 ). Computation results in: c0 :=



α1 10



,

c1 =



α+1 1 1 0



,

and the two new generators c0 and c1 define a new hash function by: h(b1 . . . bm ) := cb1 · · · cbm ∈ G We have the following: ˘ ˘ ′ ) if and only if h(v) = h(v ′ ). = h(v Lemma 4.1 Let v, v ′ ∈ V . Then h(v) Lemma 4.1. transforms the original problem with respect to s0 , s1 into the equivalent problem of finding short collisions with respect to the new, symmetric generators c0 , c1 . This is critical in our solution. More generally, conjugating by any element t ∈ G transforms the generators and hash values but preserves collisions.

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4.3.2. The structure of palindrome collisions Since a solution must be independent of the choice of the irreducible polynomial q(x), we proceed to work in SL2 (F2 [x]). Accordingly, matrices C0 , C1 ∈ SL2 (F2 [x] are defined with polynomial entries as follows: C0 :=



x1 10



,

C1 =



x+1 1 1 0



,

and a new hash function H : V −→ SL2 (F2 [x]) is defined by: H(b1 . . . bm ) := Cb1 · · · Cbm

∈ SL2 (F2 [x])

We further have:   Lemma 4.2 Let v ∈ V be a palindrome, and write H(v) = ac db . Then b = c, i. e., H(v) is symmetric. Moreover, deg(a) = |v|, and max{deg(b), deg(d)} ≤ |v|. 2×2

Now, define function ρ : V −→ F2 [x]

by:

ρ(v) := H(0v0) + H(1v1) For a given irreducible polynomial q(x) , ρ(v) ≡ ( 00 00 ) mod q(x) if and only if h(0v0) = h(1v1) is a collision in SL2 (F2 [x]/(q(x))). Lemma 4.3 If v ∈ V is a palindrome of length |v|, then ρ(v) = ( aa a0 ), where a ∈ F2 [x] has degree |v|. Moreover, a is the upper left entry of H(v). Lemma 4.4 If v ∈ V is a palindrome of even length, then H(v) = a, b, d ∈ F2 [x].



a2 b b d2



for some

Proof. If u ∈ V denote by ur the reversal of u. Let v = uur for some u ∈ V . The proof is by induction on |u|. When |u| = 0 the hash H(uur ) is the identity matrix and the statement holds trivially. Suppose now we extend a string u of given length by one bit, yielding a palindrome r βvβ = (βu)(u  β ∈ {0, 1}. By the induction hypothesis we have that H(v) =  2 β) with r a b H(uu ) = b d2 , so that: H(βvβ) = Cβ



a2 b b d2



Cβ =



(x + β)2 a2 + d2 (x + β)a2 + b (x + β)a2 + b a2



Consequently, both diagonal entries of H(βvβ) are squares, and the result follows. ✷ Combining Lemmas 4.3 and 4.4 yields:

I. Ilić and S.S. Magliveras / Crypto Applications of Combinatorial Group Theory

Corollary 4.1 Let v ∈ V be a palindrome of even length. Then ρ(v) = 2



15

 2

a2 a a2 0

for

some a ∈ F2 [x] with deg(a) = |v|/2. In particular, the entry a is the upper left entry of H(v). Further, from the proof of Lemma 4.4. we are able to deduce the following recurrence relation: Corollary 4.2 Let bn . . . b1 b1 . . . bn ∈ V be a palindrome of length 2n. Then, for 0 ≤ i ≤ n, the square root pi of the upper left entry of H(bi . . . b1 b1 . . . bi ) is given by ⎧ ⎪ if i = 0; ⎨1, pi = x + b1 + 1, if i = 1; ⎪ ⎩ (x + bi )pi−1 + pi−2 , if 1 < i ≤ n. Now, for the given irreducible polynomial q = q(x) ∈ F2 [x] of degree n, we seek a palindrome v ∈ V of length 2n such that ρ(v) = H(0v0) + H(1v1) ≡ ( 00 00 ) (modulo q(x)) in F2 [x]. 4.3.3. Mesirov and Sweet In view of Corollaries 4.1. and 4.2. , finding such a v ∈ V can be accomplished by determining a second polynomial p(x) ∈ F2 [x] of degree n − 1 such that: 1. gcd (q(x), p(x)) = 1, 2. during the execution of the Euclidean algorithm with input (q(x), p(x)), the successive quotients are all of degree 1, 3. the degree of each remainder is only one less than the degree of the respective divisor. This will ensure a “Euclidean algorithm chain” of maximal length and adherence to the recurrence relation in Corollary 4.2. The existence of such a polynomial p(x) follows from a 1987 result by J.P. Mesirov and M.M. Sweet [8]. Proposition 4.2 [Mesirov and Sweet [8]] Given any irreducible polynomial q of degree n over F2 , there is a sequence of polynomials pn , pn−1 , . . . , p0 with pn = q and p0 = 1, such that the deg (pi ) = i, and pi ≡ pi−2 mod pi−1 . Once we know a polynomial p = pn−1 , as mentioned in Proposition 4.2. , which matches our given polynomial pn = q, the Euclidean algorithm will uniquely complete the sequence pn , pn−1 , . . . , p1 , p0 = 1. The linear quotients x + βi (i = 1, . . . , n) occurring in Euclid’s algorithm allow us to derive the bits bi of the palindrome in Corollary 4.2. This has been a brief summary of the cryptanalysis of the last variant of the

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Tillich-Zémor cryptographic hash function published in [2]. A much more comprehensive development occurs in [2] including an efficient algorithm for determining from irreducible q(x) = pn (x) the two solutions for p(x) = pn−1 (x) satisfying the MesirovSweet conditions. The paper [2] also contains the solution to all suggested challenge parameters for the Tillich-Zémor hash function.

5. Conclusions With the advent of P. Shor’s quantum algorithms for solving the traditional DLP in linear time on a quantum computer, attention has been drawn to the generalized discrete logarithm problem in non-abelian groups. In this paper we consider the family of groups P SL(2, p), p an odd prime, and expose certain bad choices for generators, for which GDLP can be easily solved. We delineate some strategies for solving the GDLP in these groups, but point out that, for these strategies, the probability of success goes to zero as p gets large. We still believe that if generators are chosen wisely, the GDLP in the P SL(2, p) will be intractable. In a related problem, we summarize the general successful attack presented in [2], which breaches the Tillich-Zémor hash function. In this segment, we give no proofs for the structure lemmas, and no details in the final solution based on a theorem of Mesirov and Sweet [8].

References [1] Wieb Bosma, John Cannon, and Catherine Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput., 24(3-4):235-265, 1997 [2] Markus Grassl, Ivana Ili´c, Spyros Magliveras, Rainer Steinwandt. Cryptanalysis of the Tillich-Zémor hash function. To appear in the Journal of Cryptology, 2010. Cryptology ePrint Archive: Report 2009/376, 2009. Available at: http://eprint.iacr.org/2009/376 [3] Derek Holt, Bettina Eick, Eamonn A. O’Brien. Handbook of computational group theory. Chapman & Hall/CRC Press, Boca Raton, 2005. [4] Ivana Ili´c and Spyros S. Magliveras. Weak discrete logarithms in non-abelian groups, to appear in the Journal of Combinatorial Math. and Comb. Computing (JCMCC), 2009. [5] Lee C. Klingler, Spyros S. Magliveras, Fred Richman, Michal Sramka. Discrete logarithms for finite groups. Computing 85, (2009), pp. 3–19. [6] P. Nguyen, New Trends in Cryptology, European project STORK - “Strategic Roadmap for Crypto" (IST2002-38273). [7] Alfred Menezes, Paul C. van Oorschot, Scott A. Vanstone. Handbook of Applied Cryptography, CRC Press, 1996. [8] Jill P. Mesirov and Melvin M. Sweet. Continued Fraction Expansions of Rational Expressions with Irreducible Denominators in Characteristic 2. Journal of Number Theory 27 pp. 144–148, 1987. [9] P. W. Shor. Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J. on Computing, 26(5), pp. 1484-1509, 1997. [10] Michal Sramka. New Results in Group Theoretic Cryptology. Ph.D. Thesis, Florida Atlantic University, Boca Raton, FL 2006. [11] Douglas R. Stinson. Cryptography: Theory and Practice, 2nd ed, CRC Press, New York, NY, 2002. [12] Michio Suzuki. Group Theory I. Springer-Verlag, New York, 1982. [13] Jean-Piere Tillich and Gilles Zémor. Hashing with SL2 . LNCS 839, Advances in Cryptology – CRYPTO ’94, pp. 40–49, 1994.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-17

17

Generating rooted trees of m nodes uniformly at random Kenneth Matheis and Spyros S. Magliveras CCIS, Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431 [email protected], [email protected] Abstract. A rooted tree is an ordinary tree with an equivalence condition: two trees are the same if and only if one can be transformed into the other by reordering subtrees. In this paper, we construct a bijection and use it to generate rooted trees (or forests) of any specified nodecount m uniformly at random. As an application, Raddum and Semaev [6] Raddum and Semaev propose a technique to solve systems of polynomial equations over F2 as occurring in algebraic attacks on block ciphers. This approach is known as MRHS. In [3] Geiselmann, Matheis, and Steinwandt propose an ASIC hardware design to implement MRHS, and they show that the use of ASICs seems to enable significant performance gains over a software implementation of MRHS. What hasn’t been asserted is the total time complexity of their platform, though individual components’ runtimes are provided. If one supposes that deletions in MRHS occur as rooted trees generated uniformly at random, then one application of the proposed algorithm would be to contribute to such a time complexity; experiments are generated to provide statistical averages of key quantities. Keywords. rooted tree, rooted forest, uniform random generation, genetic programming, MRHS, PET SNAKE

Introduction We view a rooted tree as an equivalence class of ordinary trees, where two trees are equivalent if one can be transformed into the other by re-ordering subtrees [7]. Similarly, we view a rooted forest as an equivalence class of forests, where two rooted forests are equivalent if one can be transformed into the other by re-ordering the rooted trees. Alternately, we may consider a rooted forest as nothing more than the subtrees of a rooted tree (of one node more) whose root is hidden. The idea of a rooted tree has been around since 1875 [2] when countings for smaller nodecounts have been computed. Since then there have been a few proposals constructing bijections between all rooted trees and N. For each m ∈ N, we define Tm to be the set of rooted trees of m nodes, and Fm to be the set of rooted forests of m nodes. Our contribution is an implicit construction of a bijection between Fm and Z|Fm | . Such a bijection can then be used to generate a rooted forest of m nodes uniformly at random. As an immediate application to cryptography, we note that some statistics generated from these trees can be used to help calculate a time estimate for PET SNAKE. Now, In

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K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

[6] Raddum and Semaev proposed a technique known as MRHS (Multiple Right Hand Sides) to handle polynomial systems of equations over F2 . This algorithm is particulary well-suited for describing systems of equations for an algebraic key recovery attack against common block ciphers such as AES or DES, but a complete time estimate of it was not forthcoming. Later, the hardware platform PET SNAKE [3] was designed to implement MRHS attacks in hardware, but PET SNAKE’s time estimate is also hard to calculate. The statistics mentioned can help contribute to such a time estimate. Related Work It has been established that we can map rooted trees to natural numbers [5] and that there is a rooted tree for every natural number [4]. If one relaxes the equivalence condition and merely examines arbitrary trees, then a uniform random generation algorithm is known [1] by modeling them using a context-free grammar for use in a genetic algorithm. However, it is not clear that it is possible to create rooted trees using a context-free grammar, so we do not use this algorithm. We instead develop a different algorithm which, as it happens, shares some features with the one in [1]. Further, much information about rooted trees is available in [8, sequence A000081], and some of those facts will be used in this paper. Structure of the Paper We first discuss the construction of the bijection between Fm and Z|Fm | . Once this is established, we review the relevant details about MRHS and show how Fm is related to the processing of PET SNAKE (notably the deletion count therein). Finally, we generate some statistics based on Fm for m ≤ 1000 and relate those to time estimate processing for PET SNAKE.

1. Generating Rooted Forests Uniformly at Random We begin with some notation: we define the natural numbers N to be {1, 2, 3, . . . }, the whole numbers W to be N∪{0}, and for each n ∈ N, we define segn to be {1, 2, . . . , n}. In order to generate a rooted forest of m nodes uniformly at random, we first construct some data tables dynamically (so that no unneeded space is allocated), and then we perform many lookups on those tables. We view a rooted forest of m nodes as being constructed by a collection of r rooted for some r ∈ segm , with respective nonincreasing node counts c1 , trees a1 , a2 , . . . , ar , c2 , . . . , cr such that ci = m. We then construct sequences of counts bi such that b11 , b12 , . . . , b1s1 are the s1 counts starting with c1 that are equal to c1 , and b21 , b22 , . . . , b2s2 are the s2 counts starting with c1+s1 that are equal to c1+s1 , and so on, and we suppose there are d such sequences. This breaks up the counts into subsequences of equal-valued terms. For example, if the counts c were 9, 8, 8, 8, 7, 7, 6, 4, 3, 3, 3, 3, 2, 1, 1, 1, then b1 has one term (namely 9), b2 has three terms (all of which are 8), b3 has two terms (both of which are 7), b4 has one term (namely 6), and so on, ending with b8 having three terms (all of which are 1) and d = 8.

K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

19

Since we envision the trees Tk (for any k ∈ N) as being ordered, for each i ∈ segd we must count the number of ordered arrangements of si trees in Tbi1 . Call this number  Bi . We then calculate the number of rooted forests with this count sequence as Bi . In order to correlate a number in Z|Fm | to a forest in Fm , we must have a way to obtain the number of forests of subtrees with any nonincreasing count sequence. As one might imagine, this is done recursively using the building blocks described below. 1.1. Setup The setup phase of the algorithm consists of building three tables. First, for each i ∈ segm , |Ti | is calculated using the recurrence formula ⎡⎛ ⎞ ⎤ i−1 1  ⎣⎝  |Ti | = d · |Td |⎠ |Ti−k |⎦ i−1 k=1

d|k

with |T1 | = 1 [8]. This takes O(i2 ) time for each i, totalling a time of O(m3 ). Then an m × m table R called the runtable is created. Its purpose is to store forest counts in the following way: for any two i, j ∈ segm , Rij is the number of sequences u : segj → Ti such that u1 ≤ u2 ≤ · · · ≤ uj ; in other words, it is the number of nondecreasing j-length sequences of i-node rooted trees. (Note that we have not mentioned how to order the trees Ti , but certainly one exists. Indeed, a side effect of this process 1−1 is to construct the bijections fm : Z|Fm | −−→ Fm , which in turn constructs the bijeconto

1−1

tions tm : Z|Tm | −−→ Tm , so for two trees p and q, p ≤ q if and only if the index that onto constructs p is less than or equal to the index that constructs q. Since R simply concerns itself with the number of sequences u, and not the individual sequences themselves, we run into no difficulty constructing R.) To calculate these values, we take advantage of the following theorem: Theorem 1 (∀n ∈ N)(∀k ∈ W)



n  i+k i=1

k+1

=



 n+k+1 k+2

To prove this, simply use induction on “n”. Five lines are all that are necessary.  We now make an observation about sequences of finite length. (To be clear, we make no claim about the originality of Theorems 1 and 2, but absent appropriate references, proofs are provided to justify their correctness.) Theorem 2 For each i, j ∈ N, let Sij be the number of nondecreasing sequences u : segj → segi . Then (∀j ∈ N)(∀i ∈ N) Sij =



! i+j−1 . j

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K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

To prove this, we proceed by induction on “j”. ϕ(1): Let i ∈ N. Then Si1 is the number of nondecreasing sequences u : seg  1 → segi . But there are only i such things, one for each choice of u1 . Hence, Si1 = i = 1i = i+1−1 . Thus, ϕ(1). 1 " #  Let k ∈ N and assume ϕ(k): (∀i ∈ N) Sik = i+k−1 . k ϕ(k + 1): Let i ∈ N. Consider Si(k+1) . These are all the nondecreasing sequences u : segk+1 → segi . Now let us consider the possibilities for u1 . If u1 = 1, then the remaining terms comprise a k-length nondecreasing sequence to segi . But the number of such sequences is just Sik . Now, if u1 = 2, the remaining terms comprise a k-length sequence to segi − 1. But the number of such sequences is the same as the number of klength sequences to segi−1 (just subtract 1 from each term), which is S(i−1)k . Similarly, for each v ∈ segi , if u1 = v, then the remaining terms comprise a k-length sequence to segi − segv−1 , whose count is the same as the count of k-length sequences to segi−(v−1) (by subtracting v − 1 from each term), which is S(i−v+1)k . Hence, Si(k+1) = Sik + S(i−1)k + · · · + S1k =

i 

Svk

v=1

=

i  v+k−1 v=1

k

i+k k+1

i + (k + 1) − 1 . = k+1

=



by Ind. Hyp. by Theorem 1

Thus, ϕ(k + 1). The rest follows by the Principle of Mathematical Induction and routine steps.  Since, for each j ∈ segm , Rij is the number of nondecreasing sequences from segj to Ti , this is the same as the number of nondecreasing sequences from segj to seg|Ti | ,   by Theorem 2. Hence, we build the R table by populating it with this which is |Ti |+j−1 j binomial coefficient for each i ∈ segm and j ∈ segm such that j ≤ ⌊ mi ⌋; j is restricted in this way since, for any choice of tree size i in an m-node forest, you can only have at most ⌊ mi ⌋ such trees. As a side effect, we see that |Ti | is stored in Ri1 by this process. As a point of interest, note that we had to use this binomial simplification when populating the R table. Otherwise, since |Ti | is asymptotically 0.4399 · 2.9558i · i−3/2 [8, sequence A000081], asking the computer to perform the sum listed in the proof of Theorem 2 would become infeasible very quickly. We construct two more tables, the two-dimensional partable denoted P , and the three-dimensional table lentable denoted L. For each i, j ∈ segm , Pij is the number of rooted forests of i nodes whose first tree has j nodes. It could be that the first few trees have j nodes, so we keep track of this using the lentable: Lijk is the number of rooted forests of i nodes whose first k trees have j nodes. To calculate Lijk , we use

K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

Lijk =

⎧ ⎨ Rjk

⎩ j−1

q=1

Then,

21

if i − jk = 0 P(i−jk)q

otherwise

⌊i⌋

Pij =

j 

Lijk .

k=1

Finally, we recognize that |Fm | = |Tm+1 | gives us no intuitive breakdown of all the counts, but |Fm | =

m 

Pmj

j=1

does. Note that, though we concern ourselves with how a given number of nodes m breaks down into each partition of m, this setup prevents us from having to loop through each partition of m, which also would be infeasible very quickly. We remark that the storage for R is O(m2 ) but is significantly less than m2 since, for each i ∈ segm , we only populate Rij when j ≤ ⌊ mi ⌋. Further, for similar reasons the storage for L is O(m3 ), but is significantly less than m3 . 1.2. Teardown In order to generate a forest in Fm uniformly at random, we first generate a number r in Z|Fm | (called an index) uniformly at random. Then, we go through the process of whittling down r by successively discovering which count sequence to use for that forest, and which indices to use for each tree of that forest. (Such data collectively is called a decomposition of the index r.) After the decomposition is constructed, we recur on each tree size of the decomposition, noting that if the ith tree has ci nodes, it can be viewed as a forest (of its subtrees) of ci − 1 nodes, the root itself being one node. The recursion terminates when we are faced with generating a forest of one node with index zero, at which point we return a leaf. 1.2.1. Composing Decompositions For any forest of n nodes whose first tree can have as many as h nodes (called the head size), composing a decomposition is itself a recursive process which relies on three algorithms which we call PTCOORDS, LENREM , and RUNCOORDS . The process produces two vectors, sizes and idxs. PTCOORDS identifies which column of Pn that r is in (say it’s the jth) and reduces the index and provides a new head. LENREM identifies which tower of Lnj that the reduced index is in (say it’s the kth) and produces a re-reduced index, remainder index, and a remaining node count for subsequent trees. RUNCOORDS converts the re-reduced

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K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

index into a sequence of indices for each of the k trees. We then recur on the remaining node count, the new head minus 1, and the remainder index, and we append a sequence of k entries of j to the front of the result’s sizes, and also the sequence of indices to the front of the result’s idxs. This process is started with a call to DECOMP(m, m, r). Algorithm 1 DECOMP Require: A nodecount n, a head h, an index r ∈ Z|Fn | . 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12:

set sizes and idxs to be empty lists if n ≤ 0 or h < 1 then return (sizes, idxs) else if h > n then h←n end if (r′ , h′ ) ← PTCOORDS(n, h, r) (k, n′ , r′′ , x) ← LENREM(n, h′ , r′ ) f rontidxs ← RUNCOORDS(h′ , r′′ , k) set f rontsizes to be a list of k copies of h′ (backsizes, backidxs) ← DECOMP(n′ , h′ − 1, x) return (append(f rontsizes, backsizes), append(f rontidxs, backidxs))

Algorithm 2 PTCOORDS Require: A nodecount n, a head h, an index r ∈ Z|Fn | . 1: 2: 3: 4: 5: 6:

r′ ← r, h′ ← h while Pnh′ ≤ r′ do r′ ← r′ − Pnh′ h′ ← h′ − 1 end while return (r′ , h′ )

LENREM sends two of its outputs to RUNCOORDS, which uses a binary search to determine what the indices should be for each of the k trees of nodesize h′ , based on ′′ ′′ re-reduced  index r . This approach is needed since r is an index into one of the the |Th′ |+k−1 nondecreasing sequences from segk to seg|Th′ | , but this quantity is a sum k (as per Theorem 1), so we have to figure out where r′′ is in that sum without examining |Th′ | − 2 individual binomial coefficients, as |Th′ | can get very large. (Indeed, this is the part that is significantly different than the uniform random generation algorithm in [1].) We remark in passing that, after the Setup phase, building a rooted forest of m nodes corresponding to an index r takes slightly more than O(m) time but definitely within O(m2 ) time.

K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

23

2. Application to MRHS and PET SNAKE Now that we have a reliable method to generate rooted forests uniformly at random, one application would be to compute relevant statistics from them to help predict PET SNAKE’s run time. We recall the relevant facts about MRHS and PET SNAKE. MRHS operates on a collection of pairs of matrices called symbols, and one phase of its processing is called the Agreement Phase, where each symbol must be agreed to each other symbol. Sometimes the act of agreeing a pair of symbols induces a deletion in one (or both) symbols; other times, nothing changes. If a deletion occurs, then the process starts over: each symbol must be re-agreed to each other symbol. This continues until no (more) deletions are detected, at which point   the symbols are said to be pairwise agreed. Hence, for a body of n symbols, at least n2 agreements must be performed. In software, each agreement must be performed one at a time. PET SNAKE is a hardware design employ  ing lots of processors, and it uses them to perform half of the n2 agreements simultaneously. If no deletion is detected, it then performs half of the remaining agreements simultaneously. And so on. Since some deletions cannot occur until other deletions occur first, we choose to model the deletions as a collection of rooted trees. In each tree, each node symbolizes a deletion after two symbols are agreed, and each child of a node symbolizes deletions that can now occur as a result of the parent node’s deletions taking place. In the beginning of an agreement phase, it is certainly possible that many deletions do not depend on each other, so these deletions are the roots of the trees in this collection. We observe that the order of subtrees of a given node is irrelevant; it does not matter which subtree is the first subtree, which is the second, and so on; hence the choice of a rooted forest is appropriate. We notice that at any stage, PET SNAKE will perform half of the agreements necessary simultaneously, so at any point, about half of the deletions that can be performed will be performed on average. Now, if a deletion gets performed, then that deletion’s children will then be available to be deleted. Examining the consequences for the model, we see that only the roots of the trees in the forest are available for deletion, so when such a deletion is performed, the corresponding root must be eliminated. This, however, means that that root’s children are now roots in the forest. This operation of deleting a root and promoting its children we call a lift. Algorithm 3 LENREM Require: A nodecount n, a new head h′ , a reduced index r′ . 1: 2: 3: 4: 5: 6: 7: 8: 9: 10:

k ← ⌊n/h′ ⌋ while Lnh′ k ≤ r′ do r′ ← r′ − Lnh′ k k ←k−1 end while n′ ← n − (k · h′ ) c ← ⌊Lnh′ k /Rh′ k ⌋ r′′ ← ⌊r′ /c⌋ x ← r′ mod c return (k, n′ , r′′ , x)

24

K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

Algorithm 4 RUNCOORDS Require: A new head h′ , a re-reduced index r′′ , a length k. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: 19: 20: 21: 22: 23: 24:

set idxs to be an empty list top ← Rh′ 1 , prev ← 0 for t ∈ {k, k − 1, . . . , 2} by −1 do i ← top +  1, j ←1, mid ← ⌊(i + j)/2⌋ ′′ , pen ← total − 1 − r total ← top+t−1 t f ound ← false while not f ound do    > pen, c2 ← pen ≥ mid+t−2 c1 ← mid+t−1 t t f ound ← (c1 and c2 ) if not f ound then if c1 then i ← mid else j ← mid end if mid ← ⌊(i + j)/2⌋ end if end while prev ← top − mid + prev insert prev onto the back of the list  idxs  ) top ← mid, r′′ ← r′′ − (total − mid+t−1 t end for insert r′′ + prev onto the back of the list idxs return idxs

Hence, about half of the roots are lifted in a given stage. (Such an action we will refer to as a parallel lift.) The agreement phase is not complete until all the nodes in the forest are eliminated. To get a handle on time estimates, it is pertinent to ask how many roots exist at a given time, and how many times to we expect to perform parallel lifts until the forest is eliminated. Since we do not have theoretical answers to these questions, we assume that the m deletions in an agreement phase occur as a forest of m nodes chosen uniformly at random. With this assumption, we design an experiment as follows: for various m ≤ 1000, we perform the Experimental Procedure (see Figure 1) several times (say, s times). Throughout each procedure run, we count the number of roots that the forest has (once before each parallel lift) so as to calculate the average when the forest is eliminated, and we also count the number of times we have to parallel lift. Once the number of parallel lifts and the average number of roots are calculated, we do it again for the same forest. This is repeated s times. Once these s procedures are complete, we choose another rooted forest of m nodes uniformly at random and perform the procedure again. We construct t such forests (each giving rise to s procedures), and a global average of number of parallel lifts required and number of roots appearing at any point are calculated. This procedure was performed for s = t = 1000 and m ∈ {50, 100, 150, . . . , 1000} and the results are summarized in Table 1.

K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

25

• Construct a rooted forest of m nodes uniformly at random. • While it is nonempty, ∗ take note of the number of roots of the forest, ∗ uniformly at random choose half of the roots, and ∗ lift them from the forest. • Calculate the average number of roots the forest had. Figure 1. Experimental Procedure

Table 1. Experimental Procedure Results (s = t = 1000) m

Avg parallel lifts

Avg roots

m

Avg parallel lifts

Avg roots

50

25.4741

3.9869

550

100.283

11.4762

100

38.7107

5.3268

600

105.187

11.9226

150

49.2455

6.330

650

109.466

12.4351

200

56.9224

7.3193

700

112.619

13.01

250

65.1864

7.9930

750

119.717

13.1435

300

71.7676

8.7246

800

123.128

13.6412

350

78.5635

9.3096

850

125.295

14.2402

400

83.6236

10.0201

900

129.423

14.5746

450

89.7707

10.4778

950

133.577

14.9439

500

94.8623

11.0328

1000

135.625

15.4812

If we multiply the average roots by the average parallel lifts and plot this result for all twenty pairs, we discover that the plot forms a near-straight line of slope approximately 40 19 . This isn’t too surprising, since in each parallel lift we eliminate about half the roots, and the roots multiplied by the parallel lifts (if we eliminated every root per lift) should give us the total number of nodes in the forest. Further, if we multiply the number of 9 roots by itself and plot this, we get a near-straight line of slope approximately 38 . From these two observations, we propose the following: Proposition of m nodes chosen uniformly at random will have, on $ 1 A rooted forest √ 9 average, 38 m ≈ 0.4866 m roots on average as its corresponding set of deletions get deleted through an agreement phase. Further, the number of parallel lifts required to eliminate such a forest is on average √ √ 40 approximately 19 m/0.4866 m ≈ 4.3264 m. These estimates can be used in conjunction with estimates of how many deletions to expect per agreement phase to help predict the runtime of PET SNAKE. As a point of interest, if we choose not to lift half of the roots, but instead all of them, we can use a similar procedure to determine the average depth and the average number of nodes per depth for these trees.

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K. Matheis and S.S. Magliveras / Generating Rooted Trees of m Nodes Uniformly at Random

3. Conclusion We have provided a way to implicitly construct bijections between Tm and Z|Tm | , and between Fm and Z|Fm | with reasonable time and space consumption, for any m ∈ N, and we hope that this proves useful in many environments. One such environment is in the realm of cryptography, where we aid in the construction of a time estimate for a hardware platform implementing an algebraic attack on block ciphers. Another might be in genetic programming to create initial trees corresponding to non-context-free grammars.

References [1] W. Böhm and A. Geyer-Schulz, Exact Uniform Initialization for Genetic Programming, Foundations of Genetic Algorithms IV (1997), 379–403. [2] A. Cayley, On the Analytical Forms called Trees, American Journal of Mathematics 4 (1881), 266–268. [3] W. Geiselmann and K. Matheis and R. Steinwandt, PET SNAKE: A Special Purpose Architecture to Implement an Algebraic Attack in Hardware. Cryptology ePrint Archive, Report 2009/222 (2009), available at http://eprint.iacr.org/2009/222. [4] F. Göbel, On a 1-1 Correspondence between Rooted Trees and Natural Numbers, Journal of Combinatorial Theory B 29 (1980), 141–143. [5] D. Matula, A Natural Rooted Tree Enumeration by Prime Factorization, SIAM Review 10 (1968), 273. [6] H. Raddum and I. Semaev, Solving Multiple Right Hand Sides Linear Equations, Designs, Codes and Cryptography 49 (2008), 147–160. [7] F. Ruskey, Information on Rooted Trees. The Combinatorial Object Server, University of Victoria, Canada (2003), available at http://www.theory.cs.uvic.ca/~cos/inf/tree/ RootedTree.html. [8] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences. AT & T Research Labs (2009), available at http://www.research.att.com/~njas/sequences/.

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-27

On Jacobsthal Binary Sequences Spyros S. Magliveras a , Tran van Trung b and Wandi Wei a CCIS, Department of Math. Sciences, Florida Atlantic University, Boca Raton, FL 33431, USA e-mail: [email protected], [email protected] b Institute for Experimental Mathematics, University of Duisburg-Essen, Essen, Germany e-mail: [email protected] a

Abstract. Let Σ = {0, 1} be the binary alphabet, and A = {0, 01, 11} be the set of three strings 0, 01, 11 over Σ. Let A∗ denote the Kleene closure of A, Z0 the set of nonnegative integers, and Z+ the set of positive integers. A sequence in A∗ is called a Jacobsthal binary sequence. Let J(n) denote the set of Jacobsthal binary sequences of length n. For k ∈ Z+ , {s1 , s2 , . . . , sk } ⊂ Z0 , and n − 1 ≥ s1 > s2 > . . . > sk ≥ 0, let J1 (n; s1 , s2 , . . . , sk ) denote the subset J1 (n; s1 , s2 , . . . , sk ) = {an−1 an−2 . . . a1 a0 ∈ J(n) : asi = 1 (1 ≤ i ≤ k)}, of J(n), and let N1 (n; s1 , s2 , . . . , sk ) = |J1 (n; s1 , s2 , . . . , sk )|. When k = 1, a formula for N1 (n; s) has been derived recently. In this paper we consider the general case of N1 (n; s1 , s2 , . . . , sk ), and study some other special types of Jacobsthal binary sequences. Some identities involving these numbers are also given. Keywords. Jacobsthal numbers, combinatorial identities, combinatorial enumeration

Introduction Let Σ = {0, 1} be the binary alphabet, and A = {0, 01, 11} the set of three strings 0, 01, 11 over Σ. Let A∗ denote the Kleene closure of A, Z0 the set of nonnegative integers, and Z+ the set of positive integers. A sequence in A∗ is called a Jacobsthal binary sequence. Let J(n) denote the set of Jacobsthal binary sequences of length n and let |J(n)| denote the cardinality of J(n). The Jacobsthal numbers are defined by the recursion Jn = Jn−1 + 2Jn−2 ,

n>2

(1)

together with the initial values J0 = J1 = 1.

(2)

Note that some other authors use the initial values J0 = 0, J1 = 1 instead. Using the initial values in (2), a known result can be stated more conveniently as

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S.S. Magliveras et al. / On Jacobsthal Binary Sequences

|J(n)| = Jn .

(3)

Jn is also called the nth Jacobsthal number. For convenience, we also define Jm = 0, ∀m ∈ Z, m < 0.

(4)

Based on (4), we state an obvious fact and a known result as a lemma for easy reference. Lemma 1 The recursion (1) can be extended as Jt = Jt−1 + 2Jt−2 ,

t ∈ Z, t = 0.

The value of Jn (n ∈ Z0 ) can be computed by Jn =

1 n+1 + (−1)n ), (2 3

n ∈ Z0 .

(5)

The Jacobsthal numbers have applications in such areas as tiling, graph matching, alternating sign matrices, etc. ([1,2,4,5]). Let k ∈ Z+ , {s1 , s2 , . . . , sk−1 , sk } ⊂ Z0 ; n − 1 ≥ s1 > s2 > . . . > sk ≥ 0.

(6)

Let J1 (n; s1 , s2 , . . . , sk ) denote the following subset of J(n): J1 (n; s1 , s2 , . . . , sk ) = {an−1 an−2 . . . a1 a0 ∈ J(n) : asi = 1 (1 ≤ i ≤ k)}, i.e., the subset of Jacobsthal binary sequences that have the digit 1 at each of the sth i (1 ≤ i ≤ k) positions from the right. Let N1 (n; s1 , s2 , . . . , sk ) = |J1 (n; s1 , s2 , . . . , sk )|. R. Grimaldi[4] considers the case where k = 1, establishing a recursion for N1 (n; s1 ) and then deriving the following formula: 1 N1 (n; s) = (2n + (−1)n + (−1)n−s 2s ) 3 2s = Jn − (2n−s + (−1)n−s−1 ). 3

(7) (8)

For the general case, finding a formula for N1 (n; s1 , s2 , . . . , sk ) by using a recursion seems extremely difficult. In this article we employ a different approach to dealing with this problem, namely, considering the following dual problem of N1 (n; s1 , s2 , . . . , sk ). Let r ∈ Z+ , {t1 , t2 , . . . , tr−1 , tr } ⊂ Z0 , n − 1 ≥ t1 > t2 > . . . > tr ≥ 0. Let J0 (n; t1 , t2 , . . . , tr ) denote the following subset of J(n):

(9)

S.S. Magliveras et al. / On Jacobsthal Binary Sequences

29

J0 (n; t1 , t2 , . . . , tr ) = {an−1 an−2 . . . a1 a0 ∈ J(n) : ati = 0 (1 ≤ i ≤ r)}, i.e., the subset of Jacobsthal binary sequences that have the digit 0 at each of the tth i (1 ≤ i ≤ r) positions from the right. Let N0 (n; t1 , t2 , . . . , tr ) = |J0 (n; t1 , t2 , . . . , tr )|. In the next section we present characterizations of the sets J(n) and J0 (n; t1 , t2 , . . . , tr ). Based on them, some combinatorial identities involving Jn , N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) are derived in Section 3. From these identities, formulas for N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) are obtained in the last section.

1. Characterizations of the sets J(n) and J0 (n; t1 , t2 , . . . , tr )

For easy reference we state a trivial fact, that is Lemma 2 For any i, j ∈ Z+ , J(i)||J(j) ⊆ J(i + j), where J(i)||J(j) = {a||b : a ∈ J(i), b ∈ J(j) and  stands for the concatenation operation on strings. We now characterize the set J(n). We need Lemma 3 Let l ∈ Z+ . The string α of the 0-digit followed by l−1 1-digits is a Jacobsthal binary string of length l. Proof. If l = 2m + 1 for some m ∈ Z0 , the l − 1 = 2m 1-digits in α can be regarded as m copies of the string 11. Since both strings 11, 0 ∈ A, we know α ∈ A. If l = 2m for some m ∈ Z0 , the last l − 2 = 2m − 2 1-digits in α can be regarded as m − 1 copies of the string 11. Since both string 11, 01 ∈ A, we know α ∈ A. ✷ Theorem 1 For any n ∈ Z+ , a binary sequence of length n is in J(n) if and only if it is an all-1 sequence of even length or its first 0-digit from the left is preceded by an all-1 subsequence of even length. Proof. Since the string 1 ∈ A but the string 11 ∈ A, the all-1 sequence of length n is in J(n) if and only if n is even. Therefore, in what follows we only need to consider the case in which the sequence an−1 an−2 . . . a1 a0 has at least one 0-digit. Let an−i be the first 0-digit from the left. Then an−1 = an−2 = . . . = an−(i−1) = 1. Since the two strings 1, 10 ∈ A, in order for an−1 an−2 . . . a1 a0 to be in J(n), the subsequence an−1 an−2 . . . an−(i−1) has to be formed by copies of the element 11 ∈ A. This is impossible when i − 1 is odd. We now prove that when i − 1 is even, the sequence an−1 an−2 . . . a1 a0 is in J(n) by induction on the number, say u, of 0-digits in the sequence. For the case where u = 1, let ai = 0,. By Lemma 3, the subsequence ai ai−1 . . . a1 a0 ∈ J(i + 1). Recalling that

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30

an−1 an−2 . . . ai+1 ∈ J(n − i − 1) we know an−1 an−2 . . . a1 a0 ∈ J(n) by Lemma 2. This establishes the induction basis. For the inductive step, suppose that u > 1 and the conclusion is true for any sequence having exactly u−1 0-digits. Let al be the first 0-digit from the right in a sequence having u 0-digits. By Lemma 3, we know al al−1 . . . a0 = 011 . . . 1 . . . a0 ∈ J(l + 1). By the induction hypothesis, an−1 an−2 . . . al+1 ∈ J(n − l − 1). Therefore, an−1 an−2 . . . a1 a0 ∈ J(n) by Lemma 2. This completes the induction. ✷ From this theorem, one can obtain the known formula (5) for |J(n)|. Corollary 1 |J(n)| =

2n+1 + (−1)n , 3

Proof. Let J(n, i) denote the set of such Jacobsthal binary sequences that have their first 0-digit at the (2i + 1)st position from the left, and Δn the set consisting of the all-1 sequence of length n when 2 | n, and Δn = ∅ when 2 ∤ n. Then J(n) = (

%

0≤i≤(n−1)/2

J(n, i) ) ∪ Δn

is a partition of J(n). By Theorem 1, when n = 2m (m ∈ Z+ ), we have : |J(n)| =

m−1

22m−(2i+1) + 1 =

i=0

= 2

m−1 i=0

1 2

m−1

4i + 1 = 2( 4

i=0

m

−1 3 )

4(m−i) + 1 =

+1 =

1 2

m

2n+1 +(−1)n 3

i=1

4i + 1 =

.

When n = 2m + 1 (m ∈ Z0 ), we have : |J(n)| =

m

i=0

=

22m+1−(2i+1) =

m

i=0

4i =

4m+1 −1 3

m

i=0

=

22(m−i) =

2n+1 +(−1)n 3

m

i=0

22i =

. ✷

By Theorem 1 we can give a characterization of the set J0 (n; t1 , t2 , . . . , tr ). Recall that the parameters satisfy (9): r ∈ Z+ , {t1 , t2 , . . . , tr−1 , tr } ⊂ Z0 , n − 1 > t1 > t2 > . . . > tr ≥ 0. Theorem 2 For any n ∈ Z+ , the binary sequence an−1 an−2 . . . a1 a0 of length n is in J0 (n; t1 , t2 , . . . , tr ) if and only if the subsequence an−1 an−2 . . . at1 +1 is in J(n−1−t1 ) and ati = 0 (1 ≤ i ≤ r).

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Proof. Let aj be the first 0-digit from the left. Then j ≥ t1 . By Theorem 1, an−1 an−2 . . . a1 a0 ∈ J(n) if and only if the entries before aj are all 1’s, i.e., 2|n − 1 − j, which is the necessary and sufficient condition for an−1 an−2 . . . at1 +1 to be in J(n − 1 − t1 ). ✷ It is somewhat surprising that whether an−1 an−2 . . . a1 a0 ∈ J0 (n; t1 , t2 , . . . , tr ) or not is determined only by the subsequence an−1 an−2 . . . at1 +1 and ati = 0 (1 ≤ i ≤ r), but is independent of the digits aj (0 ≤ j ≤ t1 − 1, j = ti ). Based on these theorems, some combinatorial identities involving Jn , N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) can be established, which will be presented in the next section. 2. Some Combinatorial Identities Involving Jn , N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) In this section some combinatorial identities involving Jn , N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) are proved. Applying them to obtain formulas for N0 (n; t1 , t2 , . . . , tr ) and N1 (n; s1 , s2 , . . . , sk ) will be the task of the next section. We need a simple lemma : Lemma 4 For any n ∈ Z0 , 2n = 3Jn−1 + (−1)n . Proof. Recalling that J−1 = 0 (cf. (4)), we know that the statement is true when n = 0. When n ∈ Z+ , the statement is equivalent to (5). ✷ We can now state the following Theorem 3 N0 (n; t1 , t2 , . . . , tr ) = [3Jt1 −r + (−1)t1 −r+1 ]Jn−t1 −1

(10)

N0 (n; t1 , t2 , . . . , tr ) = Jn−r + (−1)n−t1 −1 Jt1 −r

(11)

Proof. By Theorem 2, for a sequence an−1 an−2 . . . a1 a0 in J0 (n; t1 , t2 , . . . , tr ), there are |J(n − t1 − 1)| = Jn−t1 −1 many choices for the subsequences an−1 an−2 . . . at1 +1 . For each of these choices, there are two choices for each of the digits aj (0 ≤ j ≤ t1 − 1, j = t2 , t3 , . . . , tr ). Noting that atj = 0 (1 ≤ j ≤ r), we have N0 (n; t1 , t2 , . . . , tr ) = |J(n − t1 − 1)| · 2t1 +1−r = Jn−t1 −1 2t1 −r+1 .

By Lemma 4, 2t1 −r+1 = 3Jt1 −r + (−1)t1 −r+1 .

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Therefore, N0 (n; t1 , t2 , . . . , tr ) = Jn−t1 −1 [3Jt1 −r + (−1)t1 −r+1 ], which is (10). Similarly, we can also write N0 (n; t1 , t2 , . . . , tr ) = = Jn−t1 −1 2t1 −r+1 1 = [2n−t1 + (−1)n−t1 −1 ]2t1 −r+1 3 1 = [2n−r+1 + (−1)n−t1 −1 2t1 −r+1 ] 3 1 = {3Jn−r + (−1)n−r+1 + (−1)n−t1 −1 [3Jt1 −r + (−1)t1 −r+1 ]} 3 = Jn−r + (−1)n−t1 −1 Jt1 −r , which proves (11). ✷ From this theorem, an identity can be immediately derived. Corollary 2 We have the identity [3Jt1 −r + (−1)t1 −r+1 ]Jn−t1 −1 = Jn−r + (−1)n−t1 −1 Jt1 −r . This identity can also be checked by using (5). Let us look at the cases r = 1 and r = 2. Corollary 3 If n − 1 ≥ u ≥ 0, then N0 (n; u) = [3Ju−1 + (−1)u ]Jn−u−1 n−u−1

N0 (n; u) = Jn−1 + (−1)

Ju−1

(12) (13)

Example 1 From (13) and J0 = J1 = 1, J2 = 3, we have N0 (1; 0) = J0 + (−1)0 J−1 = 1, N0 (2; 0) = J1 + (−1)1 J−1 = 1, N0 (2; 1) = J1 + (−1)0 J0 = 2, N0 (3; 0) = J2 + (−1)2 J−1 = 3, N0 (3; 1) = J2 + (−1)1 J0 = 2, N0 (3; 2) = J2 + (−1)0 J1 = 4. The corresponding subsets of J(n) are J0 (1; 0) = {0}, J0 (2; 0) = {00}, J0 (2; 1) = {00, 01}. J0 (3; 0) = {000, 010, 110}, J0 (3; 1) = {000, 001}, J0 (3; 2) = {000, 001, 010, 011}.

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33

Corollary 4 If n − 1 ≥ u ≥ 0, then [3Ju−1 + (−1)u ]Jn−u−1 = Jn−1 + (−1)n−u−1 Ju−1 . For N1 (n; s1 , s2 , . . . , sk ), we have Theorem 4 Suppose that s1 , s2 , . . . , sk satisfy (6). Then N1 (n; s1 , s2 , . . . , sk ) = Jn +



r 1≤r≤k (−1)





k−i 1≤i≤k−r+1 r−1



[Jn−r + (−1)n−si −1 Jsi −r ].

Proof. First of all, for any 1 ≤ r ≤ k, by (11) we have : 

1≤i1 4. Proof. See [66] for the details. 3.1.1. Elliptic subcovers Let j1 and j2 denote the j-invariants of the elliptic curves E1 and E2 from Lemma 2. The invariants j1 and j2 are the roots of the quadratic j 2 + 256

(2u3 − 54u2 + 9uv − v 2 + 27v) (u2 + 9u − 3v) =0 j + 65536 2 2 (u + 18u − 4v − 27) (u + 18u − 4v − 27)2 (12)

3.1.2. Isomorphic elliptic subcovers The elliptic curves E1 and E2 are isomorphic when equation (12) has a double root. The discriminant of the quadratic is zero for (v 2 − 4u3 )(v − 9u + 27) = 0

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Remark 3. From lemma 2, v 2 = 4u3 if and only if Aut(C) ∼ = D4 . So for C such that Aut(C) ∼ = D4 , E1 is isomorphic to E2 . It is easily checked that z1 and z2 = z0 z1 are conjugate when G ∼ = D4 . So they fix isomorphic subfields. If v = 9(u − 3) then the locus of these curves is given by, 4i51 − 9i41 + 73728i21 i3 − 150994944i23 = 0 289i31 − 729i21 + 54i1 i2 − i22 = 0

(13)

∼ For (u, v) = ( 49 , − 27 4 ) the curve has Aut(C) = D4 and for (u, v) = (137, 1206) it ∼ has Aut(C) = D6 . All other curves with v = 9(u − 3) belong to the general case, so Aut(C) ∼ = V4 . The j-invariants of elliptic curves are j1 = j2 = 256(9 − u). Thus, these genus 2 curves are parameterized by the j-invariant of the elliptic subcover. Remark 4. This embeds the moduli space M1 into M2 in a functorial way. 3.2. Isogenous degree 2 elliptic subfields In this section we study pairs of degree 2 elliptic subfields of K which are 2 or 3-isogenous. We denote by Φn (x, y) the n-th modular polynomial (see Blake et al. [9] for the formal definitions. Two elliptic curves with j-invariants j1 and j2 are n-isogenous if and only if Φn (j1 , j2 ) = 0. In the next section we will see how such modular polynomials can be generalized for higher genus. 3.2.1. 3-Isogeny Suppose E1 and E2 are 3-isogenous. Then, from equation (12) and Φ3 (j1 , j2 ) = 0 we eliminate j1 and j2 . Then, (4v − u2 + 110u − 1125) · g1 (u, v) · g2 (u, v) = 0

(14)

where g1 and g2 are given in [66]. Thus, there is a isogeny of degree 3 between E1 and E2 if and only if u and v satisfy equation (14). The vanishing of the first factor is equivalent to G ∼ = D6 . So, if Aut(C) ∼ D then E and E are isogenous of degree 3. = 6 1 2 3.2.2. 2-Isogeny Below we give the modular 2-polynomial. Φ2 = x3 − x2 y 2 + y 3 + 1488xy(x + y) + 40773375xy − 162000(x2 − y 2 )+ 8748000000(x + y) − 157464000000000

(15)

Suppose E1 and E2 are isogenous of degree 2. Substituting j1 and j2 in Φ2 we get f1 (u, v) · f2 (u, v) = 0 where f1 and f2 are displayed in [65]

(16)

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68

3.2.3. Other isogenies between elliptic subcovers If Aut(C) ∼ = D4 , then z1 and z2 are in the same conjugacy class. There are again two conjugacy classes of elliptic involutions in Aut(C). Thus, there are two degree 2 elliptic subfields (up to isomorphism) of K. One of them is determined by double root j of the equation (12), for v 2 − 4u3 = 0. Next, we determine the j-invariant j ′ of the other degree 2 elliptic subfield and see how it is related to j.

E1

n C @PPPP @@ PPP nnn~~ n n @@ PPP nn ~~~ n n PPP @@ n ~ n ~ n PPP ~  n n wn ' ′ /o /o /o E2 o / /o /o E ′ E1 2

√ √ If v 2 −4u3 = 0 then Aut(C) ∼ = V4 and P = {±1, ± a, ± b}. Then, s1 = a+ a1 +1 = 1 1 s2 . Involutions of C are τ1 : X → −X, τ2 : X → X , τ3 : X → − X . Since τ1 and τ3 fix no points of P then they lift to involutions in Aut(C). They each determine a pair of isomorphic elliptic subfields. The j-invariant of elliptic subfield fixed by τ1 is the double root of equation (12), namely j = −256

v3 v+1

To find the j-invariant of the elliptic subfields fixed by τ3 we look at the degree 2 covering φ : P1 → P1 , such that φ(±1) = 0, φ(a) = φ(− a1 ) = 1, φ(−a) = φ( a1 ) = √ a X 2 −1 −1, and φ(0) = φ(∞) = ∞. This covering is, φ(X) = a−1 X . The branch √

a points of φ are qi = ± √2ia−1 . From lemma 2 the elliptic subfields E1′ and E2′ have 2-torsion points {0, 1, −1, qi }. The j-invariants of E1′ and E2′ are

j ′ = −16

(v − 15)3 (v + 1)2

Then Φ2 (j, j ′ ) = 0, so E1 and E1′ are isogenous of degree 2. Thus, τ1 and τ3 determine degree 2 elliptic subfields which are 2-isogenous.

4. Theta functions In this section we give a brief description of the basic setup. All of this material can be found in any standard book on theta functions. Let C be a genus g ≥ 2 algebraic curve. We choose a symplectic homology basis for C, say {A1 , . . . , Ag , B1 , . . . , Bg }, such that the intersection products Ai · Aj = Bi · Bj = 0 and Ai · Bj = δij , where δij is the Kronecker+ delta. We choose a basis {wi } for the space of holomorphic 1-forms such that Ai wj = δij . The matrix "+ # O = Bi wj is the period matrix of C. The columns of the matrix [I |O] form a lattice L in Cg and the Jacobian of C is Jac (C) = Cg /L. Let Hg be the Siegel upper-half space. Then O ∈ Hg and there is an injection

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69

Mg ֒→ Hg /Sp2g (Z) =: Ag where Sp2g (Z) is the symplectic group. For any z ∈ Cg and τ ∈ Hg Riemann’s theta function is defined as θ(z, τ ) =



eπi(u

t

τ u+2ut z)

u∈Zg

where u and z are g−dimensional column vectors and the products involved in the formula are matrix products. The fact that the imaginary part of τ is positive makes the series absolutely convergent over any compact sets. Therefore, the function is analytic. The theta function is holomorphic on Cg × Hg and satisfies θ(z + u, τ ) = θ(z, τ ),

θ(z + uτ, τ ) = e−πi(u

t

τ u+2z t u)

· θ(z, τ ),

where u ∈ Zg ; see [54] for details. Any point e ∈ Jac (C) can be written !uniquely a 1 for the as e = (b, a) g , where a, b ∈ Rg . We shall use the notation [e] = O b g characteristic of e. For any a, b ∈ Q , the theta function with rational characteristics is defined as !  t t a (z, τ ) = θ eπi((u+a) τ (u+a)+2(u+a) (z+b)) . b g u∈Z

When the entries!of column vectors a and b are from the set {0, 21 }, then the a characteristics are called the half-integer characteristics. The corresponding b theta functions with rational characteristics are called theta characteristics. A scalar obtained by evaluating a theta characteristic at z = 0 is called a theta constant. Points of order n on Jac C are called the n1 -periods. Any half-integer characteristic is given by 1 1 m= m= 2 2 where Then,

mi , m′i

γ′ ∈ Z. For γ = γ ′′

!

∈



m1 m2 · · · mg m′1 m′2 · · · m′g

1 2g 2g 2 Z /Z



we define e∗ (γ) = (−1)4(γ

′ t

) γ ′′

.

θ[γ](−z, τ ) = e∗ (γ)θ[γ](z, τ ). We say that γ is an even (resp. odd) characteristic if e∗ (γ) = 1 (resp. e∗ (γ) = −1). For any curve of genus g, there are 2g−1 (2g + 1) (respectively 2g−1 (2g − 1) ) even theta functions (respectively odd theta functions). Let a be another half integer characteristic. We define m a as follows.

1 t1 t2 · · · tg ma = 2 t′1 t′2 · · · t′g

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70

where ti ≡ (mi + ai ) mod 2 and t′i ≡ (m′i + a′i ) mod 2. For the rest of this section we consider only characteristics 12 q in which each of the elements qi , qi′ is either 0 or 1. We use the following abbreviations |m| =

g 

mi m′i ,

|m, a| =

i=1

g  i=1

(m′i ai − mi a′i ),



Pg ′ m = eπi j=1 mj aj . a

|m, a, b| = |a, b| + |b, m| + |m, a|,

The set of all half integer characteristics forms a group Γ which has 22g elements. We say that two half integer characteristics m and a are syzygetic (resp., azygetic) if |m, a| ≡ 0 mod 2 (resp., |m, a| ≡ 1 mod 2) and three half integer characteristics m, a, and b are syzygetic if |m, a, b| ≡ 0 mod 2. A G¨ opel group G is a group of 2r half integer characteristics where r ≤ g such that every two characteristics are syzygetic. The elements of the group G are formed by the sums of r fundamental characteristics; see [2, pg. 489] for details. Obviously, a G¨ opel group of order 2r is isomorphic to C2r . The proof of the following lemma can be found on [2, pg. 490]. Lemma 3. The number of different G¨ opel groups which have 2r characteristics is (22g − 1)(22g−2 − 1) · · · (22g−2r+2 − 1) (2r − 1)(2r−1 − 1) · · · (2 − 1) If G is a G¨ opel group with 2r elements, then it has 22g−r cosets. The cosets are called G¨ opel systems and denoted by aG, a ∈ Γ. Any three characteristics of a G¨ opel system are syzygetic. We can find a set of characteristics called a basis of the G¨ opel system which derives all its 2r characteristics by taking only the combinations of any odd number of characteristics of the basis. Lemma 4. Let g ≥ 1 be a fixed integer, r be as defined above and σ = g − r. Then opel systems which consist of even characteristics only there are 2σ−1 (2σ + 1) G¨ and there are 2σ−1 (2σ − 1) G¨ opel systems which consist of odd characteristics. The other 22σ (2r − 1) G¨ opel systems consist as many odd characteristics as even characteristics. Proof. The proof can be found on [2, pg. 492]. Corollary 1. When r = g we have only one (resp., 0) G¨ opel system which consists of even (resp., odd) characteristics. Proposition 3. The following statements are true. 2

2

θ [a]θ [ah] =

1 2g−1

 e

πi|ae|

e



h 2 θ [e]θ2 [eh] ae

(17)

T. Shaska and L. Beshaj / The Arithmetic of Genus Two Curves

θ4 [a] + eπi|a,h| θ4 [ah] =

1 2g−1

 e

eπi|ae| {θ4 [e] + eπi|a,h| θ4 [eh]}

71

(18)

where θ[e] is the theta constant corresponding to the characteristic e, a and h are any half integer characteristics and e is an even characteristic such that |e| ≡ |eh| mod 2. There are 2 · 2g−2 (2g−1 + 1) such candidates for e. Proof. For the proof, see [2, pg. 524]. The statements given in the proposition above can be used to get identities among theta constants; see section 3. 4.1. Cyclic curves with extra automorphisms A normal cyclic curve is an algebraic curve C such that there exist a normal cyclic ¯ = G/Cm embeds as a subgroup Cm ⊳ Aut(C) such that g(C/Cm ) = 0. Then G finite subgroup of P GL(2, C). An affine equation of a birational model of a cyclic curve can be given by the following y m = f (x) =

s 

i=1

(x − αi )di , 0 < di < m.

(19)

Hyperelliptic curves are cyclic curves with m = 2. Note that when 0 < di for some i the curve is singular. A hyperelliptic curve C is a cover of order two of the projective line P1 . Let z be the generator (the hyperelliptic involution) of the Galois group Gal(C/P1 ). It is known that z is a normal subgroup of the automorphism group Aut(C). Let C −→ P1 be the degree 2 hyperelliptic projection. We can assume that infinity is a branch point. Let B := {α1 , α2 , · · · , α2g+1 } be the set of other branch points. Let S = {1, 2, · · · , 2g + 1} be the index set of B and ξ : S −→ 21 Z2g /Z2g be a map defined as follows; ! 0 · · · 0 12 0 · · · 0 ξ(2i − 1) = 1 1 2 ··· 2 0 0 ··· 0 ! 0 · · · 0 21 0 · · · 0 ξ(2i) = 1 1 1 2 ··· 2 2 0 ··· 0

th where the nonzero ! element of the first row appears in i column. We define ξ(∞) 0 ··· 0 0 . For any T ⊂ B, we can define the half-integer characteristic as to be 0 ··· 0 0

ξT =



ξ(k).

ak ∈T

Let T c denote the complement of T in B. Note that ξB ∈ Z2g . If we view ξT as an element of 12 Z2g /Z2g then ξT = ξT c . Let  denote the symmetric difference

72

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of sets, that is T  R = (T ∪ R) − (T ∩ R). It can be shown that the set of subsets of B is a group under . We have the following group isomorphism 1 {T ⊂ B | #T ≡ g + 1 mod 2}/T ∼ = Z2g /Z2g . 2   of the even For hyperelliptic curves, it is known that 2g−1 (2g + 1) − 2g+1 g theta constants are zero. The following theorem provides a condition on the characteristics in which theta characteristics become zero. The proof of the theorem can be found in [55, pg. 102]. Theorem 2. Let C be a hyperelliptic curve, with a set B of branch points. Let S be the index set as above and U be the set of all odd values of S. Then for all T ⊂ S with even cardinality, we have θ[ξT ] = 0 if and only if #(T △U ) = g + 1, where θ[ξT ] is the theta constant corresponding to the characteristics ξT . Notice also that by parity, all odd theta constants are zero. There is a formula (so called Frobenius’ theta formula) which half-integer theta characteristics for hyperelliptic curves satisfy. Lemma 5 (Frobenius). For all zi ∈ Cg , 1 ≤ i ≤ 4 such that z1 + z2 + z3 + z4 = 0 and for all bi ∈ Q2g , 1 ≤ i ≤ 4 such that b1 + b2 + b3 + b4 = 0, we have 

ǫU (j)

j∈S∪{∞}

4 

θ[bi + ξ(j)](zi ) = 0,

i=1

where for any A ⊂ B, , 1 ǫA (k) = −1

if k ∈ A otherwise

Proof. See [54, pg. 107]. A relationship between theta constants and the branch points of the hyperelliptic curve is given by Thomae’s formula. Lemma 6 (Thomae). For a non singular even half integer characteristics e corresponding to the partition of the branch points {1, 2, · · · , 2(g + 1)} = {i1 < i2 < · · · < ig+1 } ∪ {j1 < j2 < · · · < jg+1 }, we have θ[e](0; τ )8 = A



k 3 are more difficult to handle. Recently, Shaska dealt with cases n = 5, 7 in [49]. The locus of C, denoted by Ln , is an algebraic subvariety of the moduli space M2 . The space L2 was studied in Shaska/V¨ olklein [66]. The space Ln for n = 3, 5 was studied by Shaska in [63,49] were an algebraic description was given as sublocus of M2 . 5.1. Curves of genus 2 with split Jacobians Let C and E be curves of genus 2 and 1, respectively. Both are smooth, projective curves defined over k, char(k) = 0. Let  ψ : C −→ E be a covering of degree n. From the Riemann-Hurwitz formula, P ∈C (eψ (P ) − 1) = 2 where eψ (P ) is the ramification index of points P ∈ C, under ψ. Thus, we have two points of ramification index 2 or one point of ramification index 3. The two points of ramification index 2 can be in the same fiber or in different fibers. Therefore, we have the following cases of the covering ψ: Case I: There are P1 , P2 ∈ C, such that eψ (P1 ) = eψ (P2 ) = 2, ψ(P1 ) = ψ(P2 ), and ∀P ∈ C \ {P1 , P2 }, eψ (P ) = 1. Case II: There are P1 , P2 ∈ C, such that eψ (P1 ) = eψ (P2 ) = 2, ψ(P1 ) = ψ(P2 ), and ∀P ∈ C \ {P1 , P2 }, eψ (P ) = 1. Case III: There is P1 ∈ C such that eψ (P1 ) = 3, and ∀P ∈ C \ {P1 }, eψ (P ) = 1. In case I (resp. II, III) the cover ψ has 2 (resp. 1) branch points in E. Denote the hyperelliptic involution of C by w. We choose O in E such that w restricted to E is the hyperelliptic involution on E. We denote the restriction of w on E by v, v(P ) = −P . Thus, ψ ◦ w = v ◦ ψ. E[2] denotes the group of 2-torsion points of the elliptic curve E, which are the points fixed by v. The proof of the following two lemmas is straightforward and will be omitted. Lemma 10. a) If Q ∈ E, then ∀P ∈ ψ −1 (Q), w(P ) ∈ ψ −1 (−Q). b) For all P ∈ C, eψ (P ) = eψ (w(P )).

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Let W be the set of points in C fixed by w. Every curve of genus 2 is given, up to isomorphism, by a binary sextic, so there are 6 points fixed by the hyperelliptic involution w, namely the Weierstrass points of C. The following lemma determines the distribution of the Weierstrass points in fibers of 2-torsion points. Lemma 11. The following hold: 1. ψ(W ) ⊂ E[2] 2. If n is an odd number then i) ψ(W ) = E[2] ii) If Q ∈ E[2] then #(ψ −1 (Q) ∩ W ) = 1 mod (2) 3. If n is an even number then for all Q ∈ E[2], #(ψ −1 (Q)∩W ) = 0 mod (2)

Let πC : C −→ P1 and πE : E −→ P1 be the natural degree 2 projections. The hyperelliptic involution permutes the points in the fibers of πC and πE . The ramified points of πC , πE are respectively points in W and E[2] and their ramification index is 2. There is φ : P1 −→ P1 such that the diagram commutes. π

C C −→ P1 ψ↓ ↓φ πE P1 E −→

(27)

Next, we will determine the ramification of induced coverings φ : P1 −→ P1 . First we fix some notation. For a given branch point we will denote the ramification of points in its fiber as follows. Any point P of ramification index m is denoted by (m). If there are k such points then we write (m)k . We omit writing symbols for unramified points, in other words (1)k will not be written. Ramification data between two branch points will be separated by commas. We denote by πE (E[2]) = {q1 , . . . , q4 } and πC (W ) = {w1 , . . . , w6 }. 5.2. Maximal coverings ψ : C −→ E Let ψ1 : C −→ E1 be a covering of degree n from a curve of genus 2 to an elliptic curve. The covering ψ1 : C −→ E1 is called a maximal covering if it does not factor through a nontrivial isogeny. A map of algebraic curves f : X → Y induces maps between their Jacobians f ∗ : JY → JX and f∗ : JX → JY . When f is maximal then f ∗ is injective and ker(f∗ ) is connected, see [61] for details. Let ψ1 : C −→ E1 be a covering as above which is maximal. Then ψ ∗ 1 : E1 → JC is injective and the kernel of ψ1,∗ : JC → E1 is an elliptic curve which we denote by E2 . For a fixed Weierstrass point P ∈ C, we can embed C to its Jacobian via iP : C −→ JC x → [(x) − (P )]

(28)

Let g : E2 → JC be the natural embedding of E2 in JC , then there exists g∗ : JC → E2 . Define ψ2 = g∗ ◦ iP : C → E2 . So we have the following exact sequence g

ψ1,∗

0 → E2 −→ JC −→ E1 → 0

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The dual sequence is also exact ψ∗

g∗

1 0 → E1 −→ JC −→ E2 → 0

If deg(ψ1 ) is an odd number then the maximal covering ψ2 : C → E2 is unique. If the cover ψ1 : C −→ E1 is given, and therefore φ1 , we want to determine ψ2 : C −→ E2 and φ2 . The study of the relation between the ramification structures of φ1 and φ2 provides information in this direction. The following lemma (see answers this question for the set of Weierstrass points W = {P1 , . . . , P6 } of C when the degree of the cover is odd. Lemma 12. Let ψ1 : C −→ E1 , be maximal of degree n. Then, the map ψ2 : C → E2 is a maximal covering of degree n. Moreover, i) if n is odd and Oi ∈ Ei [2], i = 1, 2 are the places such that #(ψi−1 (Oi ) ∩ W ) = 3, then ψ1−1 (O1 ) ∩ W and ψ2−1 (O2 ) ∩ W form a disjoint union of W.   ii) if n is even and Q ∈ E[2], then # ψ −1 (Q) = 0 or 2.

The above lemma says that if ψ is maximal of even degree then the corresponding induced covering can have only type I ramification. 5.3. The locus of genus two curves with (n, n) split Jacobians

Two covers f : X → P1 and f ′ : X ′ → P1 are called weakly equivalent if there is a homeomorphism h : X → X ′ and an analytic automorphism g of P1 (i.e., a Moebius transformation) such that g ◦ f = f ′ ◦ h. The covers f and f ′ are called equivalent if the above holds with g = 1. Consider a cover f : X → P1 of degree n, with branch points p1 , ..., pr ∈ P1 . Pick p ∈ P1 \ {p1 , ..., pr }, and choose loops γi around pi such that γ1 , ..., γr is a standard generating system of the fundamental group Γ := π1 (P1 \ {p1 , ..., pr }, p), in particular, we have γ1 · · · γr = 1. Such a system γ1 , ..., γr is called a homotopy basis of P1 \ {p1 , ..., pr }. The group Γ acts on the fiber f −1 (p) by path lifting, inducing a transitive subgroup G of the symmetric group Sn (determined by f up to conjugacy in Sn ). It is called the monodromy group of f . The images of γ1 , ..., γr in Sn form a tuple of permutations σ = (σ1 , ..., σr ) called a tuple of branch cycles of f . We say a cover f : X → P1 of degree n is of type σ if it has σ as tuple of branch cycles relative to some homotopy basis of P1 minus the branch points of f . Let Hσ be the set of weak equivalence classes of covers of type σ. The Hurwitz space Hσ carries a natural structure of an quasiprojective variety. We have Hσ = Hτ if and only if the tuples σ, τ are in the same braid orbit Oτ = Oσ . In the case of the covers φ : P1 → P1 from above, the corresponding braid orbit consists of all tuples in Sn whose cycle type matches the ramification structure of φ.

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5.3.1. Humbert surfaces Let A2 denote the moduli space of principally polarized Abelian surfaces. It is well known that A2 is the quotient of the Siegel upper half space H2 of symmetric complex 2 × 2 matrices with positive definite imaginary part by the action of the symplectic group Sp4 (Z). Let Δ be a fixed positive integer and N∆ be the set of matrices τ=



z1 z2 z2 z 3



∈ H2

such that there exist nonzero integers a, b, c, d, e with the following properties: az1 + bz2 + cz3 + d(z22 − z1 z3 ) + e = 0 Δ = b2 − 4ac − 4de

(29)

The Humbert surface H∆ of discriminant Δ is called the image of N∆ under the canonical map H2 → A2 := Sp4 (Z) \ H2 , see [36,10,53] for details. It is known that H∆ = ∅ if and only if Δ > 0 and Δ ≡ 0 or 1 mod 4. Humbert (1900) studied the zero loci in Eq. (29) and discovered certain relations between points in these spaces and certain plane configurations of six lines; see [36] for more details. For a genus 2 curve C defined over C, [C] belongs to Ln if and only if the isomorphism class [JC ] ∈ A2 of its (principally polarized) Jacobian JC belongs to the Humbert surface Hn2 , viewed as a subset of the moduli space A2 of principally polarized Abelian surfaces; see [53, Theorem 1, p. 125] for the proof of this statement. In [53] is shown that there is a one to one correspondence between the points in Ln and points in Hn2 . Thus, we have the map: Hσ −→ Ln −→ Hn2 ([f ], (p1 , . . . , pr ) → [C] → [JC ]

(30)

In particular, every point in Hn2 can be represented by an element of H2 of the form τ=



1 n 1 n z2

z1



,

z1 , z2 ∈ H.

There have been many attempts to explicitly describe these Humbert surfaces. For some small discriminant this has been done in [66], [63], [49]. Geometric characterizations of such spaces for Δ = 4, 8, 9, and 12 were given by Humbert (1900) in [36] and for Δ = 13, 16, 17, 20, 21 by Birkenhake/Wilhelm.

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5.4. Genus 2 curves with degree 3 elliptic subcovers This case was studied in detail in[63]. The main theorem was: Theorem 4. Let K be a genus 2 field and e3 (K) the number of Aut(K/k)-classes of elliptic subfields of K of degree 3. Then; i) e3 (K) = 0, 1, 2, or 4 ii) e3 (K) ≥ 1 if and only if the classical invariants of K satisfy the irreducible equation F (J2 , J4 , J6 , J10 ) = 0 displayed in [63, Appendix A]. There are exactly two genus 2 curves (up to isomorphism) with e3 (K) = 4. The case e3 (K) = 1 (resp., 2) occurs for a 1-dimensional (resp., 2-dimensional) family of genus 2 curves, see [63].

Figure 3. Shaska’s surface as graphed in [4]

A geometrical interpretation of the Shaska’s surface (the space L3 ) and its singular locus can be found in [4]. Lemma 13. Let K be a genus 2 field and E an elliptic subfield of degree 3. i) Then K = k(X, Y ) such that Y 2 = (4X 3 + b2 X 2 + 2bX + 1)(X 3 + aX 2 + bX + 1)

(31)

for a, b ∈ k such that (4a3 + 27 − 18ab − a2 b2 + 4b3 )(b3 − 27) = 0

(32)

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The roots of the first (resp. second) cubic correspond to W (1) (K, E), (resp. W (2) (K, E)) in the coordinates X, Y , (see Theorem 3). ii) E = k(U, V ) where U=

X3

X2 + aX 2 + bX + 1

and V 2 = U3 + 2

ab2 − 6a2 + 9b 2 12a − b2 4 U + U− R R R

(33)

where R = 4a3 + 27 − 18ab − a2 b2 + 4b3 = 0. iii) Define u := ab,

v := b3

Let K ′ be a genus 2 field and E ′ ⊂ K ′ a degree 3 elliptic subfield. Let a′ , b′ be the associated parameters as above and u′ := a′ b′ , v = (b′ )3 . Then, there is a k-isomorphism K → K ′ mapping E → E ′ if and only if exists a third root of unity ξ ∈ k with a′ = ξa and b′ = ξ 2 b. If b = 0 then such ξ exists if and only if v = v ′ and u = u′ . iv) The classical invariants of K satisfy equation [63, Appendix A]. Let F (X) := X 3 + aX 2 + bX + 1 G(X) := 4X 3 + b2 X 2 + 2bX + 1

(34)

Denote by R = 4a3 + 27 − 18ab − a2 b2 + 4b3 the resultant of F and G. Then we have the following lemma. Lemma 14. Let a, b ∈ k satisfy equation (32). Then equation (31) defines a genus 2 field K = k(X, Y ). It has elliptic subfields of degree 3, Ei = k(Ui , Vi ), i = 1, 2, where Ui , and Vi are as follows: U1 =

where

X2 , F (X)

V1 = Y

8 (X − s)2 (X − t) > > > > G(X) > > > > < (3X − a) U2 = > 3(4X 3 + 1) > > > > > > (bX + 3)2 > : b2 G(X)

X 3 − bX − 2 F (X)2

if

b(b3 − 4ba + 9) = 0

if

b=0

if

(b3 − 4ba + 9) = 0

(35)

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3 s=− , b

87

2

t=

b3

3a − b − 4ab + 9

8√ 27 − b3 Y > > ((4ab − 8 − b3 )X 3 − (b2 − 4ab)X 2 + bX + 1) > > 2 > G(X) > > > > < 8X 3 − 4aX 2 − 1 V2 = Y > (4X 3 + 1)2 > > > > > > 8√ Y > > : (bX 3 + 9X 2 + b2 X + b) b b G(X)

if

b(b3 − 4ba + 9) = 0

if

b=0

if

(b3 − 4ba + 9) = 0

(36)

5.5. Elliptic subcovers We express the j-invariants ji of the elliptic subfields Ei of K, from Lemma 14, in terms of u and v as follows: j1 = 16v

(vu2 + 216u2 − 126vu − 972u + 12v 2 + 405v)3 (v − 27)3 (4v 2 + 27v + 4u3 − 18vu − vu2 )2

j2 = −256

(37)

(u2 − 3v)3 2 v(4v + 27v + 4u3 − 18vu − vu2 )

where v = 0, 27. Remark 5. The automorphism ν ∈ Galk(u,v)/k(r1 ,r2 ) permutes the elliptic subfields. One can easily check that: ν(j1 ) = j2 ,

ν(j2 ) = j1

Lemma 15. The j-invariants of the elliptic subfields satisfy the following quadratic equations over k(r1 , r2 ); j 2 − T j + N = 0,

(38)

where T, N are given in [63]. 5.5.1. Isomorphic Elliptic Subfields Suppose that E1 ∼ = E2 . Then, j1 = j2 implies that 8v 3 + 27v 2 − 54uv 2 − u2 v 2 + 108u2 v + 4u3 v − 108u3 = 0

(39)

or 324v 4 u2 − 5832v 4 u + 37908v 4 − 314928v 3 u − 81v 3 u4 + 255879v 3 + 30618v 3 u2 − 864v 3 u3 − 6377292uv 2 + 8503056v 2 − 324u5 v 2 + 2125764u2 v 2 − 215784u3 v 2 4 2

6 2

3

5

6

4

6

+ 14580u v + 16u v + 78732u v + 8748u v − 864u v − 157464u v + 11664u = 0

(40)

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The former equation is the condition that det(Jac(θ)) = 0. The expressions of i1 , i2 , i3 we can express u as a rational function in i1 , i2 , and v. This is displayed in [63, Appendix B]. Also, [k(v) : k(i1 )] = 8 and [k(v) : k(i2 )] = 12. Eliminating v we get a curve in i1 and i2 which has degree 8 and 12 respectively. Thus, k(u, v) = k(i1 , i2 ). Hence, e3 (K) = 1 for any K such that the associated u and v satisfy the equation; see [63] for details. 5.5.2. The Degenerate Case We assume now that one of the extensions K/Ei from Lemma 14 is degenerate, i.e. has only one branch point. The following lemma determines a relation between j1 and j2 . Lemma 16. Suppose that K/E2 has only one branch point. Then, 729j1 j2 − (j2 − 432)3 = 0 For details of the proof see Shaska [63]. Making the substitution T = −27j1 we get j1 = F2 (T ) =

(T + 16)3 T

where F2 (T ) is the Fricke polynomial of level 2. If both K/E1 and K/E2 are degenerate then ,

729j1 j2 − (j1 − 432)3 = 0 729j1 j2 − (j2 − 432)3 = 0

(41)

There are 7 solutions to the above system. Three of which give isomorphic elliptic curves j1 = j2 = 1728,

j1 = j2 =

√ 1 (297 ± 81 −15) 2

The other 4 solutions are given by: ,

729j1 j2 − (j1 − 432)3 = 0 j12 + j22 − 1296(j1 + j2 ) + j1 j2 + 559872 = 0

(42)

5.6. Further remarks If e3 (C) ≥ 1 then the automorphism group of C is one of the following: Z2 , V4 , D4 , or D6 . Moreover; there are exactly 6 curves C ∈ L3 with automorphism group D4 and six curves C ∈ L3 with automorphism group D6 . They are listed in [62] where rational points of such curves are found. Genus 2 curves with degree 5 elliptic subcovers are studied in [49] where a description of the space L5 is given and all its degenerate loci. The case of degree 7 is the first case when all possible degenerate loci occur.

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We have organized the results of this paper in a Maple package which determines if a genus 2 curve has degree n = 2, 3 elliptic subcovers. Further, all its elliptic subcovers are determined explicitly. We intend to implement the results for n = 5 and the degenerate cases for n = 7.

6. Modular Polynomials for genus 2 The term modular polynomial refers to pollynomials which parametrize isogenies of elliptic curves as for example those in equations (15), (14). Recentely there have been efforts to define modular polynomials for higher genus, mostly by Lauter and her collaborators as in [5]. This section is merely a quick recap of that paper with some suggestions on how to compute some of these polynomials. Let Hg = {τ ∈ M atg (C) | τ T = τ, Im(τ ) > 0} be the Siegel upper half plane. We denote with J the matrix J=



0 Ig −Ig 0



.

The symplectic group Sp(2g, Z) = {M ∈ GL(4, Z) | M JM T = J} acts on Hg , Sp (2g, Z) × Hg → Hg a b  × τ → (aτ + b)(cτ + d)− 1 c d where a, b, c, d, τ are g × g matrices. From now on we take g = 2. Let A/C be a 2-dimensional principally polarized Abelian variety, and let N ≥ 1 be a positive integer. The N -torsion A[N ] of A is, non-canonically, isomorphic to (Z/N Z)4 . The polarization on A induces a symplectic form v on the rank 4 (Z/N Z)-module A[N ]. We choose a basis for A[N ] such that v is given by the matrix  0 I  2 , −I2 0 and we let Sp(4, Z/N Z) be the subgroup of the matrix group GL(4, Z/N Z) that respects v. A subspace G ⊂ A[N ] is called isotropic if v restricts to the zero-form on G × G, and we say that A and A′ are (N, N )-isogenous if there is an isogeny A → A′ whose kernel is isotropic of order N 2 . The full congruence subgroup Γ2(N ) of level N is defined as  the kernel of the reduction map Sp(4, Z) → Sp(4, Z/N Z). Explicitly, a matrix ac db is contained

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in Γ2(N ) if and only if we have a, b ≡ I2 mod N and d, c ≡ 02 mod N . The congruence subgroup Γ2(N ) fits in an exact sequence 1 −→ Γ2(N ) −→ Sp(4, Z) −→ Sp(4, Z/N Z) −→ 1. The surjectivity is not completely trivial. The 2-dimensional analogue of the subgroup Γ0 (N ) ⊂ SL2 (Z) occurring in the equality Y0 (N ) = Γ0 (N )\Hg of Riemann surfaces is the group (2) Γ0 (N )

=

-

a b ∈ Sp(4, Z) | c ≡ 02 mod N c d

.

.

From now on, we restrict to the case N = p prime. The following lemma gives the (2) link between the group Γ0 (p) and isotropic subspaces of the p-torsion, see [5] (2)

Lemma 17. The index [Sp(4, Z) : Γ0 (p)] equals the number of 2-dimensional isotropic subspaces of the Fp -vector space F4p . Let S(p) be the set of equivalence classes of pairs (A, G), with A a 2dimensional principally polarized Abelian variety and G ⊂ A[p] a 2-dimensional isotropic subspace. Here, two pairs (A, G) and (A′ , G′ ) are said to be isomorphic if there exists an isomorphism of Abelian varieties ϕ : A → A′ with ϕ(G) = G′ . (2)

Theorem 5. The quotient space Γ0 (p)\H2 is in canonical bijection with the set S(p) via 1 1 (2) Γ0 (p)τ → (Aτ , ( , 0, 0, 0), (0, , 0, 0) ) p p where Aτ = C2 /(Z2 + Z2 τ ) is the variety associated to τ . As a quotient space, the 2-dimensional analogue of the curve Y0 (p) is (2)

(2)

Y0 (p) := Γ0 (p)\H2 . (2)

Problem 1. Let g = 2. Determine Y0 (N ). (2)

It is shown in [5] that Y0 (p) has the structure of a quasi-projective variety. Siegel defined a metric on H2 that respects the action of the symplectic (2) group. With this metric, Y0 (p) becomes a topological space. Just as in the 1dimensional case Y0 (p), it is not compact. We have this Lemma from [5] (2)

Lemma 18. i) Y0 (N ) is a quasi projective variety non compact of dimension 2. ii) The Satake compactification (2)

(2)

Y0 (N )∗ = Y0 (N ) ∪ Y0 (N ) ∪ P1 (Q) is a projective variety.

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91

For a fixed prime p we define three functions Ii : H2 → P1 (C) τ → Ii (pτ ). In [5] it is claimed that Lemma 19. If N = p is a prime then we have the following: (2)

i) C(Y0 (N )) = C(I1 , I2 , I3 ) 4 −1 ii) [k(I1 ) : k] = pp−1 . The N -th modular polynomial ΨN for i1 is defined as the minimal polynomial of Ii over k. Let the corresponding polynomials of field extensions k(I1 )/k, k(I2 )/k, k(I3 )/k be ΨN , ΩN , ΛN , respectively. They are called modular polynomials of genus 2 and level N . Problem 2. Consider the following problems: i) Compute explicitly k(I1 , I2 , I3 )/k or C(I1 , I2 , I3 ). ii) Compute ΨN , ΩN , ΛN , which are the polynomials Fj (i1 , i2 , i3 , Ij ) = 0 for j = 1, 2, 3. Let each of the polynomials above be given by some equation Ad Id1 + ... + A1 I1 + A0 = 0,

(43)

and As ∈ C(i1 , i2 , i3 ), s = 1, ..., d. Lemma 20 (Broker, Lauter 2009). The coefficients As of the Eq. 43 are rational Ns functions in i1 , i2 , i3 , so As = D for s = 1, ..., d and Ns , Ds ∈ C[i1 , i2 , i3 ]. s Let LN (i1 , i2 , i3 ) be the polynomial representing the Humbert space H2 or the space LN . For N = p prime LN | Ds for all s = 1, ..., d. 6.1. Computation of modular polynomials To compute polynomials ΨN , ΩN , ΛN the following algorithm is suggested in Dupont’s thesis, see [20]. • • • •

Compute deg Ds , deg Ns over C(i1 , i2 , i3 ). Fix β, γ ∈ Q. Take some values α1 , . . . , αr . For triples (αj , β, γ) find the genus 2 curve Cj using the Rational_Model function of the genus 2 package described in Section 7. • For the curve Cj find the corresponding τj . • Then find the coefficients of I1 , I2 , I3 for the given τj .

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In this process are needed explicit equations of LN . The method is not efficient, since computation of LN is quite difficult and much information is ’lost’ from the ideal.

Algorithm 1 Algorithm for computing the modular polynomials. Require: The number p-prime. Ensure: Modular polynomials Ψp , Ωp , Λp . 1: Pick a matrix τ ∈ H2 which depends on three parameters α1 , α2 , α3 . 2: Find the genus 2 curve C corresponding to τ . 3: Compute i1 , i2 , i3 as functions of α1 , α2 , α3 . 4: Compute pτ ∈ H2 5: Compute the genus 2 C ′ corresponding to pτ . 6: Find I1 , I2 , I3 for the curve C ′ as functions of α1 , α2 , α3 . 7: Create a system with six equations ⎧ i1 − f1 (α1 , α2 , α3 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i2 − f2 (α1 , α2 , α3 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ i3 − f3 (α1 , α2 , α3 ) = 0 ⎪ I1 − g1 (pα1 , pα2 , pα3 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ I2 − g2 (pα1 , pα2 , pα3 ) = 0 ⎪ ⎪ ⎪ ⎪ ⎩ I3 − g3 (pα1 , pα2 , pα3 ) = 0

8:

where fj , gj , are rational functions for j = 1, 2, 3. Since M2 has dimension 2 there are at most 3 parameters α1 , α2 , α3 . Eliminate α1 , α2 , α3 for the three first equations. The result are the modular polynomials Ψp , Ωp , Λp .

Such algorithm requires some elimination theory or Groebner basis argument to eliminate α1 , α2 , α3 . For details see [18].

7. A computational package for genus two curves Genus 2 curves are the most used of all hyperelliptic curves due to their application in cryptography and also best understood. The moduli space M2 of genus 2 curves is a 3-dimensional variety. To understand how to describe the moduli points of this space we need to define the invariants of binary sextics. For details on such invariants and on the genus 2 curves in general the reader can check [37], [65], [44].

i1 := 144

J4 , J22

i2 := −1728

J2 J4 − 3J6 , J23

i3 := 486

J10 , J25

(44)

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93

for J2 = 0. In the case J2 = 0 we define α1 :=

J4 · J6 , J10

α2 :=

J6 · J10 J44

(45)

to determine genus two fields with J2 = 0, J4 = 0, and J6 = 0 up to isomorphism. For a given genus 2 curve C the corresponding moduli point p = [C] is defined as ⎧ (i1 , i2 , i3 ) if J2 = 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (α1 , α2 ) if J2 = 0, J4 = 0, J6 = 0 ⎪ ⎪ ⎪ ⎨ 5 J6 p= 3 if J2 = 0, J4 = 0, J6 = 0 ⎪ ⎪ J ⎪ 10 ⎪ ⎪ ⎪ ⎪ ⎪ J5 ⎪ ⎩ 24 if J2 = 0, J6 = 0, J4 = 0 J10

Notice that the definition of α1 , α2 can be totally avoided if one uses absolute invariants with J10 in the denominator. However, the degree of such invariants is higher and therefore they are not effective computationally. We have written a Maple package which finds most of the common properties and invariants of genus two curves. While this is still work in progress, we will describe briefly some of the functions of this package. The functions in this package are: J_2, J_4, J_6, J_10, J_48, L_3_d, a_1, a_2, i_1, i_2, i_3, theta_1, theta_2, theta_3, theta_4, AutGroup, CurvDeg3EllSub_J2, CurveDeg3EllSub, Ell_Sub, LocusCurves,Aut_D4, LocusCurvesAut_D4_J2, LocusCurvesAut_D6, LocusCurvesAut_V4, Rational_Model, Kummer. Next, we will give some examples on how some of these functions work. 7.1. Automorphism groups

A list of groups that can occur as automorphism groups of hyperelliptic curves is given in [65] among many other references. The function in the package that computes the automorphism group is given by AutGroup(). The output is the automorphism group. Since there is always confusion on the terminology when describing certain groups we also display the GAP identity of the group from the SmallGroupLibrary. For a fixed group G one can compute the locus of genus g hyperelliptic curves with automorphism group G. For genus 2 this loci is well described as subvarieties of M2 . Example 1. Let y 2 = f (x) be a genus 2 curve where f := x5 + 2x3 − x. Then the function AutGroup(f,x) displays: > AutGroup(f,x);

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94

[D4 , (8, 3)] 2

Example 2. Let y = f (x) be a genus 2 curve where f := x6 + 2x3 − x. Then the function AutGroup(f,x) displays: > AutGroup(f,x); [V4 , (4, 2)] We also have implemented the functions: LocusCurvesAut_V_4(), LocusCurvesAut_D_4(), LocusCurvesAut_D4_J2(), LocusCurvesAut_D_6(), which gives equations for the locus of curves with automorphism group D4 or D6 . 7.2. Genus 2 curves with split Jacobians A genus 2 curve which has a degree n maximal map to an elliptic curve is said to have (n, n)-split Jacobian; see [62] for details. Genus 2 curves with split Jacobian are interesting in number theory, cryptography, and coding theory. We implement an algorithm which checks if a curve has (3, 3), and (5, 5)-split Jacobian. The case of (2, 2)-split Jacobian corresponds to genus 2 curves with extra involutions and therefore can be determined by the function LocusCurvesAut_V_4(). The function which determines if a genus 2 curve has (3, 3)-split Jacobian is CurvDeg3EllSub() if the curve has J2 = 0 and CurvDeg3EllSub_J_2 () otherwise; see [8]. The input of CurvDeg3EllSub() is the triple (i1 , i2 , i3 ) or the pair (α1 , α2 ) for CurvDeg3EllSub_J_2 (). If the output is 0, in both cases, this means that the corresponding curve to this moduli point has (3, 3)-split Jacobian. Below we illustrate with examples in each case. Example 3. Let y 2 = f (x) be a genus 2 curve where f := 4x6 + 9x5 + 8x4 + 10x3 + 5x2 + 3x + 1. Then, > i_1:=i_1(f,x); i_2:=i_2(f,x); i_3:=i_3(f,x); i1 :=

78741 , 100

i2 :=

53510733 , 2000

i3 :=

38435553 51200000

> CurvDeg3EllSub(i1 , i2 , i3 ); 0 This means that the above curve has a (3, 3)-split Jacobian.

√ Example 4. Let y 2 =√f (x) be a genus 2√curve where f := 4x6 + (52 6 − 119)x5 + √ (39 6 − 24)x4 + (26 6 − 54)x3 + (13 6 − 27)x2 + 3x + 1. Then, > a_1:=a_1(f,x); a_2:=a_2(f,x); 1316599234443 √ 6310855638567 6+ , 270840023 541680046 1467373119039023 −96672521239976 √ 6+ a2 := 1183208072032328121 7099248432193968726

a1 :=

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95

> CurvDeg3EllSub_J_2(a1 , a2 ) 0 This means that the curve has J2 = 0 and (3, 3)-split Jacobian. 7.3. Rational model of genus 2 curve For details on the rational model over its field of moduli see [61]. The rational model of C (if such model exists) is determined by the function Rational_Model(). √ Example 5. Let y 2 = f (x) be a genus 2 curve where f := x5 + 2x3 + x. Then, > Rational_Model(f,x); 1 x5 + x3 + x 2 Example 6. Let y 2 = f (x) be a genus 2 curve where f := 5x6 + x4 + Then,

√

2x + 1.

> Rational_Model(f,x);

6

− 365544026018739971082698131028050365165449396926201478x

5

− 606501618836700589954579317910699990585971018672445125x

4

− 369842283192872727990502041940062429271727924754392250x 3

− 32387676975314893414920003149434215247663074288356250x

2

+ 74168490079198328987047652288420271784298171220937500x + 38274648493772601723357350829541971828965732551171875x + 6501732463119213927460859571034949543087123367187500

Notice that our algorithm doesn’t always find the minimal rational model of the curve. An efficient way to do this has yet to be determined. 7.4. A different set of invariants As explained in Section 2, invariants i1 , i2 , i3 were defined that way for computational benefits. However, they make the results involve many subcases and are inconvinient at times. In the second version the the genus2 package we intend to convert all the results to the t1 , t2 , t3 invariants t1 =

J25 , J10

t2 =

J45 2 , J10

t3 =

J65 3 . J10

The other improvement of version two is that when the moduli point p is given the equation of the curve is given as the minimal equation over the minimal field of definition.

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8. Further directions Genus 2 curves have been suggested for factorization of large numbers as in [16]. In the algorithm suggested in [16] certain genus 2 curves with (2, 2) have been used. We believe that we have better candidates for selecting such curves. This is work planned to be presented in [35]. The computation of modular polynomials is also a very challenging computational problem. We have made some progress on levels p = 3, 5. Equations of the moduli spaces of genus 2 curves with (3, 3) and (5, 5)-split Jacobians computed in [63] and [49] have been fundamental in such computations. The newer version of our genus 2 package will come out soon. It has functions on equations for the Kummer surface KC , the map from KC to Jac C , and conversion of most of the equations in invariants t1 , t2 , t3 .

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Elezi, Artur, Toric fibrations and mirror symmetry. Albanian J. Math. 1 (2007), no. 4, 223–233. K. Eisentrager, K. Lauter, A CRT algorithm for constructing genus 2 curves over finite fields, to appear in Arithmetic, Geometry and Coding Theory (AGCT-10), 2005. A. Enge, Computing modular polynomials in quasi-linear time. Math. Comp. 78 (2009), no. 267, 1809–1824. Elkin, Arsen; Pries, Rachel, Hyperelliptic curves with a-number 1 in small characteristic. Albanian J. Math. 1 (2007), no. 4, 245–252. J. W. Cassels and V. E. Flynn, Prolegomena to a middlebrow arithmetic of curves of genus 2. (English summary) London Mathematical Society Lecture Note Series, 230. Cambridge University Press, Cambridge, 1996. xiv+219 pp. ISBN: 0-521-48370-0 Gashi, Qndrim R., A vanishing result for toric varieties associated with root systems. Albanian J. Math. 1 (2007), no. 4, 235–244. P. Gaudry, Fast genus 2 arithmetic based on theta functions, J. Math. Cryptol. 1 (2007), 243–265. ´ Schost, On the invariants of the quotients of the Jacobian of a curve of P. Gaudry and E. genus 2, Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (S. Bozta¸s and I. Shparlinski, eds.), Lecture Notes in Comput. Sci., vol. 2227, Springer-Verlag, 2001, pp. 373–386. P. Gaudry, R. Harley, Counting points on hyperelliptic curves over finite fields,Algorithmic Number Theory Symposium IV, Springer Lecture Notes in Computer Sience, vol. 1838, 2000, pp. 313-332. P. Gaudry, T. Houtman, D. Kohel, C. Ritzenthaler, A. Weng, The 2-adic CMmethod for genus 2 curves with applications to cryptography, Asiacrypt, Springer Lecture Notes in Computer Science, vol. 4284, 2006, pp. 114-129 P. Gaudry, E. Schost, Modular equations for hyperelliptic curves, Math, Comp,74 vol. (2005), 429-454. J. Gutierrez and T. Shaska, Hyperelliptic curves with extra involutions, LMS J. of Comput. Math., 8 (2005), 102-115. Haran, D.; Jarden, M., Regular lifting of covers over ample fields. Albanian J. Math. 1 (2007), no. 4, 179–185. R. Hidalgo, Classical Schottky uniformizations of Genus 2. A package for MATHEMATICA. Sci. Ser. A Math. Sci. (N.S.) 15 (2007), 6794. V. Hoxha and T. Shaska, Factoring large numbers by using genus two curves, (work in progress) G. Humbert Sur les fonctionnes abliennes singulires. I, II, III. J. Math. Pures Appl. serie 5, t. V, 233–350 (1899); t. VI, 279–386 (1900); t. VII, 97–123 (1901). J. Igusa, Arithmetic Variety Moduli for genus 2. Ann. of Math. (2), 72, 612-649, 1960. J. -I. Igusa, On Siegel modular forms of genus two, Amer. J. Math.84 (1962), 175-200. C. Jacobi, Review of Legendre, Th´eorie des fonctions elliptiques. Troiseme suppl´em ent. 1832. J. reine angew. Math. 8, 413-417. B. Justus, On integers with two prime factors. Albanian J. Math. 3 (2009), no. 4, 189–197. Joswig, Michael; Sturmfels, Bernd; Yu, Josephine Affine buildings and tropical convexity. Albanian J. Math. 1 (2007), no. 4, 187–211. Joyner, David; Ksir, Amy; Vogeler, Roger, Group representations on Riemann-Roch spaces of some Hurwitz curves. Albanian J. Math. 1 (2007), no. 2, 67–85 (electronic). A. Krazer, Lehrbuch der Thetafunctionen, Chelsea, New York, 1970. ¨ lklein, Invariants of binary forms , Developments V. Krishnamorthy, T. Shaska, H. Vo in Mathematics, Vol. 12, Springer 2005, pg. 101-122. A. Krazer, Lehrbuch der Thetafunctionen, Chelsea, New York, (1970). Kopeliovich, Yaacov, Modular equations of order p and theta functions. Albanian J. Math. 1 (2007), no. 4, 271–282. H. W. Lenstra, Jr., Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), 649–673. Luca, Florian; Shparlinski, Igor E., Pseudoprimes in certain linear recurrences. Albanian J. Math. 1 (2007), no. 3, 125–131 (electronic).

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T. Shaska and L. Beshaj / The Arithmetic of Genus Two Curves ¨ lklein, Genus 2 curves with degree 5 elliptic subcovers, K. Magaard, T. Shaska, H. Vo Forum. Math., vol. 16, 2, pg. 263-280, 2004. ¨ lklein, Helmut; Wiesend, Go ¨ tz, The combinatorics of degenerate Magaard, Kay; Vo covers and an application for general curves of genus 3. Albanian J. Math. 2 (2008), no. 3, 145–158. ¨ lklein, The locus of curves with K. Magaard, T. Shaska, S. Shpectorov, and H. Vo prescribed automorphism group. Communications in arithmetic fundamental groups (Kyoto, 1999/2001). S¯ urikaisekikenky¯ usho K¯ oky¯ uroku No. 1267 (2002), 112–141. J. -F. Mestre, Construction des curbes de genre 2 a partir de leurs modules, Effective Methods in Algebraic Geometry, Birkhauser, Progress in Mathematics, vol. 94, 1991, pp. 313-334. D. Mumford, The Red Book of Varieties and Schemes, Springer, 1999. D. Mumford, Tata lectures on theta. II. Jacobian theta functions and differential equations. With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. Progress in Mathematics, 43. Birkhuser Boston, Inc., Boston, MA, 1984. D. Mumford, Tata lectures on theta. I. With the assistance of C. Musili, M. Nori, E. Previato and M. Stillman. Progress in Mathematics, 28. Birkhuser Boston, Inc., Boston, MA, 1983. xiii+235 pp. N. Murabayashi, The moduli space of curves of genus two covering elliptic curves, Manuscripta Math.84 (1994), 125-133. A.Nakayashiki, On the Thomae formula for ZN curves, Publ. Res. Inst. Math. Sci., vol 33 (1997), no. 6, pg. 987–1015. Previato, E,; Shaska, T.; Wijesiri, S., Thetanulls of cyclic curves of small genus, Albanian J. Math., vol. 1, Nr. 4, 2007, 265-282. H.E. Rauch and H.M.Farkas, Theta functions with applications to Riemann surfaces, Williams and Wilkins, Baltimore, 1974. R. Sanjeewa, Automorphism groups of cyclic curves defined over finite fields of any characteristics. Albanian J. Math. 3 (2009), no. 4, 131–160. T. Shaska, Curves of genus 2 with (n, n)-decomposable Jacobians, J. Symbolic Comput. 31 (2001), no. 5, 603–617. T. Shaska, Genus 2 curves with (3,3)-split Jacobian and large automorphism group, Algorithmic Number Theory (Sydney, 2002), 6, 205-218, Lect. Not. in Comp. Sci., 2369, Springer, Berlin, 2002. T. Shaska, Genus 2 curves with degree 3 elliptic subcovers, Forum. Math., vol. 16, 2, pg. 263-280, 2004. T. Shaska, Some special families of hyperelliptic curves, J. Algebra Appl., vol 3, No. 1 (2004), 75-89. T.Shaska, Genus 2 curves covering elliptic curves, a computational approach Lect.Notes in Comp. 13 (2005) ¨ lklein, Elliptic subfields and automorphisms of genus two fields, T. Shaska and H. Vo Algebra, Arithmetic and Geometry with Applications, pg. 687 - 707, Springer (2004). T. Shaska and S. Wijesiri, Theta functions and algebraic curves with automorphisms, Algebraic Aspects of Digital Communications, pg. 193-237, NATO Advanced Study Institute, vol. 24, IOS Press, 2009. H. Shiga, On the representation of the Picard modular function by θ constants. I, II., Publ. Res. Inst. Math. Sci., vol. 24, (1988), no. 3, pg. 311–360. P. van Wamelen, Equations for the Jacobian of a hyperelliptic curve, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3083–3106.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-99

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Covering arrays and hash families Charles J. COLBOURN CIDSE, Arizona State University, P.O. Box 878809, Tempe, AZ 85287-8809, U.S.A. [email protected] Abstract. The explicit construction of covering arrays arises in many disparate applications in which factors or components interact. Despite this, current computational tools are effective only when the number of factors is small, while probabilistic methods are typically effective only when the number of factors is very large. Consequently combinatorial constructions have played, and continue to play, a significant role. Although some direct constructions from codes, Steiner systems, Hadamard matrices, and arrays over the finite field provide very useful examples, the workhorses of the combinatorial methods are the recursive constructions. There are two main classes of recursive techniques, the cut-and-paste or Roux-type constructions, and the column replacement techniques. After describing both for strength two, the focus is on column replacement techniques. In particular, constructions that use hash families to select columns from smaller covering arrays are examined, in order to understand the interplay among properties of the hash families and covering arrays needed to produce effective constructions. This leads to specializations both of hash families and covering arrays that merit further investigation. Keywords. covering array, perfect hash family, separating hash family, distributing hash family, heterogeneous hash family, interaction testing.

1. Covering Arrays Let N , k, t, and v be positive integers. Let C be an N × k array with entries from an alphabet Σ of size v; we typically take Σ = {0, . . . , v−1}. When (ν1 , . . . , νt ) is a t-tuple with νi ∈ Σ for 1 ≤ i ≤ t, (c1 , . . . , ct ) is a tuple of t column indices (ci ∈ {1, . . . , k}), and ci = cj whenever νi = νj , the t-tuple {(ci , νi ) : 1 ≤ i ≤ t} is a t-way interaction. (In this definition, fewer than t distinct columns may be involved, and so strictly speaking the interaction may not be considered to be “t-way” in certain contexts, but we find this extension convenient.) The array covers the t-way interaction {(ci , νi ) : 1 ≤ i ≤ t} if, in at least one row ρ of C, the entry in row ρ and column ci is νi for 1 ≤ i ≤ t. Array C is a covering array CA(N ; t, k, v) of strength t when every t-way interaction is covered. Applications to interaction testing, in particular to testing component-based software, have driven much recent research; see [34,35,41,43,134]. In applications in testing, columns of the array correspond to experimental factors, and the symbols in the column form values or levels for the factor. Each row specifies the values to which to set the factors for an experimental run. The array is ‘covering’ in the sense that every t-way interaction appears in at least one run. Figure 1 gives an example of a covering array with N = 13 rows, ten factors having two levels each, and strength three. Consider, for example, the 3-way interaction {(2, 0), (5, 1), (6, 1)}; it is covered in the sixth and ninth

100

C.J. Colbourn / Covering Arrays and Hash Families

  rows. (The diligent reader can check all of the 8 10 3 = 960 3-way interactions at their leisure.) 0 1 1 1 1 0 0 1 0 0 0 1 0

0 1 1 0 0 1 0 1 0 0 1 0 1

0 1 1 1 0 1 1 0 0 1 0 0 0

0 1 0 1 0 0 0 1 1 1 1 0 0

0 1 1 0 1 0 1 0 1 0 1 0 0

0 1 0 1 1 1 0 0 1 0 0 0 1

0 1 0 0 1 0 1 1 0 1 0 0 1

0 1 0 1 0 0 1 0 0 0 1 1 1

0 1 0 0 0 1 1 1 1 0 0 1 0

0 1 1 0 0 0 0 0 1 1 0 1 1

Figure 1. CA(13;3,10,2)

We denote by CAN(t, k, v) the minimum N for which a CA(N ; t, k, v) exists, because fewer rows means fewer tests to be run. Because CAN(1, k, v) = v, CAN(t, k, v) = v t when k < t, and CAN(t, k, 1) = 1, we generally assume that k ≥ t ≥ 2 and v ≥ 2. Nevertheless, the definition employed herein allows t, k, and v to be arbitrary positive integers. Our primary concern in this paper is with recursive constructions that make larger covering arrays from smaller ones by a technique of ‘column replacement’. Before narrowing to these specific topics, however, we provide some background in order to place them in context. 1.1. Applications and Equivalent Formulations Covering arrays are employed in numerous testing applications in which experimental factors interact to detect the presence of faults (see [44,75] and references therein), to detect the location of faults [58,96], to detect interactions in biological networks [115], to generate representative multiple sequence alignments of genomic data [80,111], to quantify uncertainty in measurement [85], and to learn an unknown function by nonadaptive tests [62]. In these applications, often factors have differing numbers of levels. Permitting different numbers of levels in each column leads to mixed covering arrays [56,104,116]; here we concentrate on the uniform case. For further applications, other formulations have been explored. Let C be an N × k covering array. Suppose that rows are indexed by a set R of size N . Then each column can be viewed as a partition of R into exactly v classes (M1 , . . . , Mv ); the class Mi of r ∈ R is determined by the value i appearing in row r in the chosen column. In this manner, an array gives a collection P = {R1 , R2 , . . . , Rk } of partitions of R. A family of partitions is t-qualitatively independent when for every t of the partitions /t Ri1 , . . . Rit , and for every choice of classes Mij ∈ Rij , for 1 ≤ j ≤ t, we find that j=1 Mij = ∅; this concept was pioneered by Marczewski [95] in 1948. It follows that covering arrays

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of strength t having N rows are the same as t-qualitatively independent partitions of a set of size N . Many early results were established using this vernacular, but generally we translate to the language of covering arrays. Poljak, Pultr, and Rödl [108] and Körner and Lucertini [87] discuss combinatorial problems related to qualitative independence. A (k, t)-universal set is a subset of {0, 1}k such that the projection on every t coordinates contains all 2t combinations. Hence it is a CA(t, k, 2). Naor and Naor [105] establish that (k, t)-universal sets arise as probability spaces with limited independence; indeed these have been extensively studied as ǫ-biased arrays [3,89,92,105]. Bierbrauer and Schellwatt [15] extend this framework to more than two levels per column; see also [8]. Certain binary covering arrays are equivalent to face transversals of the n-cube; see [14,81,82,72]. Binary covering arrays arise in rendezvous search on the line [68]. Certain other binary covering arrays yield “existentially closed graphs” [6,20,64,73]; the relation to covering arrays is given in [52]. The nomenclature t-surjective array for a covering array of strength t is also used, to indicate that on each t columns, every possible outcome arises. See [1,30,37,38,79,114], for example. Our goal is not to treat these many different formulations and their applications here, but they provide ample reasons for constructing covering arrays! A special case of covering arrays is of particular interest (see [77], for example). Let k, t, and v be positive integers. A v t × k array, each column of which contains v distinct symbols, is an orthogonal array OA(t, k, v) of strength t when, for every way to select t columns, each of the v t possible tuples of symbols arises in exactly one row. A key property of orthogonal arrays, not shared by covering arrays in general, is that every two distinct rows have the same symbols in at most t − 1 of the columns (for otherwise, one of the v t possible tuples of symbols would arise in at least two rows). A transversal design of order n, blocksize k, and strength s, denoted by TD(s, k, n), is a triple (V, G, B). V is a set of kn points partitioned into groups G = {G1 , . . . , Gk }, with each group of size n. The set B contains ns blocks, each of which is a subset of V of size k; each block meets each group in a single element (i.e. it is transverse to the groups), and two distinct blocks intersect in fewer than s elements. The transversal design TD(s, k, n) is equivalent to an orthogonal array OA(s, k, n) of strength s and index unity. The equivalence is straightforward. Form the TD(s, k, n) on {0, . . . , n−1}×{1, . . . , k} and let group Gi = {0, . . . , n − 1} × {i} for 1 ≤ i ≤ k. Then each block B of the TD forms a row of the OA, by placing j in column i when (j, i) ∈ B. See [50,51] for background on transversal designs, and [77] for orthogonal arrays. We mention one standard construction of orthogonal arrays that is used repeatedly. Let q be a prime power and q ≥ s ≥ 2. Over the finite field Fq , let F = {F1 , . . . , Fqs } be the set of all polynomials of degree less than s. Let A be a subset of Fq ∪ {∞}. Define a q s ×|A| array in which the entry in cell (a, j) is Fj (a) when a ∈ Fq , and is the coefficient of the term of degree s − 1 in Fj when a = ∞. The result is an OA(q s ; s, |A|, q). A TD or OA is linear if it is constructed in this way. 1.2. Explicit Determination of Covering Arrays The determination of CAN(t, k, v) has been the subject of much research; see [32,44, 75,76] for survey material. For fixed t and v, only CAN(2, k, 2) has been determined exactly [83,86,110]. In fact, an explicit construction of covering arrays with the fewest rows when t = v = 2 is given there. Beyond this, when t and v are fixed, exact numbers

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are known only for a few small values of k (see [53], for example). Therefore most effort has focussed on constructions of covering arrays that have ‘few’ rows, that is, on upper bounds for CAN(t, k, v). Asymptotic results can be used to determine the growth rate of CAN(t, k, v) for fixed t and v as a function of k (see [69,70] for t = 2 and [71] in general, for example). Nevertheless the explicit construction of covering arrays is required for many of the applications mentioned. Asymptotic results typically rely on selecting arrays at random, and showing that when the number of rows is large enough, the randomly selected array is a covering array with high probability. A simple experiment illustrates this. For 9 ≤ N ≤ 100, we generated one hundred random N × 100 arrays on three symbols, and checked, for each, whether it is a CA(N ; 2, 100, 3). Random Matrices Percent Success or Covered

100 80 60 40 20

0

20

40 60 Number of Rows

80

100

Figure 2. Random CA(N ; 2, 100, 3)s

The lower band of points in Figure 2 shows the observed probability of success. The first success occurs at N = 74, when one out of one hundred arrays meets the requirements. Thereafter the probability climbs rapidly. It has been frequently argued (and sometimes vehemently argued [9]) that in practice requiring every t-way interaction to be covered is too restrictive, and that if all but a few are, that ought to suffice. So the upper band of points in Figure 2 shows the observed percentage of the 44550 2-way interactions that are covered (on average). This gives a much more optimistic portrait of the randomly selected arrays. Once N ≥ 40, the percentage of interactions covered exceeds 99%, and for some practical purposes this may suffice. (See [74] for a general discussion.) Nevertheless, using the combinatorial methods to be described in this paper, one can produce a CA(24; 2, 100, 3) [56]. For N = 24 in this example, a randomly selected array covers only about 93.3% of the interactions. Hence we argue that for construction of small arrays, naive random methods are not the appropriate solution. Let us turn to explicit constructions. Orthogonal arrays provide a number of specific examples [77], as do Hadamard matrices [52], cyclotomic classes in the finite fields [48], error-correcting codes [118], and Steiner systems [53]. In [45,120], the structure of the finite field leads to a projection technique that reduces the number of symbols while increasing the number of columns.

C.J. Colbourn / Covering Arrays and Hash Families

103

Computational methods produce many more arrays. For example, simulated annealing [42,53], tabu search [107], backtracking [133], integer programming [118], and constraint satisfaction [78] have proved successful for strengths up to four, but have limited application to larger strengths at present. Local optimization can often reduce the number of rows required [106]. Compact representations of covering arrays as ‘permutation vectors’ can be used to extend heuristic search methods to these larger strengths [117,132], but they restrict to a subset of the admissible parameter sets. Assuming the presence of certain automorphisms also can reduce the difficulty of computational search [31,45,102]. Nevertheless, for strength at least five, greedy methods [25,26,29,66,91,93] and random methods [90] are often the only ones for which competitive computational results can be obtained in a ‘reasonable’ amount of time. Consequently, an extensive amount of research has concentrated on recursive methods. Roux [112] developed a simple but effective doubling construction for binary covering arrays of strength three. Roux-type constructions operate by juxtaposing copies of smaller covering arrays, sometimes with smaller strength. Such constructions have been explored for strength three [32,42,57], strength four [57,75], strength five [98], and arbitrary strength [75,97,98]. For strengths three and four, these constructions often yield the smallest known covering arrays for v ≤ 25 and k ≤ 10000 [47]. For strengths five and larger, they appear to be less effective at present. A further class of recursive constructions instead selects columns from a smaller covering array, using the easy observation that any t columns from a covering array of strength t cover all t-way interactions. These form the main focus of the paper, and the remaining sections develop these constructions in detail. 1.3. Some Further Definitions and Notation Two rows of a CA(N ; t, k, v) are disjoint if they do not agree in any column. In general, some t-way interactions may be covered more than once. Now consider each row r of a CA(N ; t, k, v); for every subset C of t columns, let T be the t-way interaction that is covered in row r and the columns of C. If T is not covered in any other row of the array, each of the cells {(r, c) : c ∈ C} is necessary. All cells that are not necessary in this way are flexible. If we ignore a flexible cell (r, c) in the computation of coverage, all t-way interactions remain covered – this is the meaning of flexibility here. By convention, when flexible cell (r, c) is to be ignored, we place the entry ⋆ (“don’t care”) in cell (r, c). For example, in Figure 1, one can verify that each of the 92 22 3-way interactions containing the “0” in the (1,1) cell appears more than once, and hence this position is flexible – and can be changed to ⋆. Formally this modifies the definition of CA as follows: An N × k array, each cell of which contains one of v distinct symbols or a different symbol ⋆, is a covering array CA(N ; t, k, v) of strength t when, for every way to select t columns, each of the v t possible tuples arises in at least one row. In general, one cannot simply convert all flexible cells to ⋆, because two flexible cells can each rely on the value in the other for its flexibility. Nevertheless, one can repeatedly choose any one flexible cell to convert to ⋆, and then recalculate the flexible cells for this modified CA, until none remain. The profile (d1 , . . . , dk ) of an N × k array is a k-tuple in which the entry di is the number of ⋆ entries in the ith column. A single covering array can often admit many different profiles, by filling the ⋆ cells and changing a (possibly different) set of flexible cells to ⋆. A profile (d1 , . . . , dk ) dominates profile (e1 , . . . , ek ) when di ≥ ei for 1 ≤ i ≤ k, and we write (d1 , . . . , dk ) ≥ (e1 , . . . , ek ) in this case.

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A row r = (r1 , . . . , rk ) of a CA(N ; t, k, v) is duplicated if there is another row s = (s1 , . . . , sk ) for which for each 1 ≤ i ≤ k, we have ri = si , ri = ⋆, or si = ⋆. It is irrelevant if it covers no t-way interaction not covered in another row, or equivalently if every cell in the row is flexible. Duplicated rows are irrelevant, but not all irrelevant rows need be duplicated. One irrelevant row can be removed without reducing coverage; indeed this can be iterated until no row is irrelevant. For example, in Figure 1, one can verify that every cell in the first row is flexible, and hence that this row can be removed without reducing coverage. A row is constant if, for some symbol ν, every entry in the row is either ν or ⋆. A row is pure constant if it is constant and contains no ⋆. For example, in Figure 1, the first two rows are pure constant rows. Because symbols within each column can be permuted independently, one has: Observation 1.1 If a CA(N ; t, k, v) exists having ρ rows that are pairwise disjoint, there is a CA(N ; t, k, v) having ρ constant rows. These can without loss of generality be assumed to be on any ρ of the v symbols. In a standardized CA(N ; t, k, v) the first row is constant. Any CA(N ; t, k, v) can be rewritten by choosing a column, and applying an arbitrary permutation to the symbols in the column. Observation 1.2 If a CA(N ; t, k, v) exists, then a standardized CA(N ; t, k, v) exists. 2. Products of Strength Two Covering Arrays We begin with the easiest case, that of strength t = 2. We consider a simple product construction, and explore methods to refine it. 2.1. Direct Products We start with the simplest direct product, which appears in different vernacular in [109] and [36]; it is also the essence of the block recursive construction from [121]. Theorem 2.1 [109] When a CA(N ; 2, k, v) and a CA(M ; 2, ℓ, v) both exist, a CA(N + M ; 2, kℓ, v) also exists. Proof. Let A = (aij ) be a CA(N ; 2, k, v) and let B = (bij ) be a CA(M ; 2, ℓ, v). Form an (N + M ) × kℓ array C = (ci,j ) = A ⊗ B by setting ci,(f −1)k+g = ai,g for 1 ≤ i ≤ N , 1 ≤ f ≤ ℓ, and 1 ≤ g ≤ k. Then set cN +i,(f −1)k+g = bi,f for 1 ≤ i ≤ M , 1 ≤ f ≤ ℓ, and 1 ≤ g ≤ k. In essence, k copies of B = (bij ) are being appended to ℓ copies of A = (aij ) as shown in Figure 3. Because two different columns of C arise either from different columns of A or from two different columns of B, the result is a CA(N + M ; 2, kℓ, v).  An extension of this simple concatenation to exploit “don’t care” cells and constant rows is considered in [56]; we extend it further here. We suppose that a factor with v values always takes on values from {0, . . . , v − 1}, and hence the corresponding column of the array contains only these symbols, and possibly ⋆. A CA(M ; 2, ℓ, v) B and a CA(M ′ ; 2, ℓ′ , v) B′ are (L, r)-compatible if for every 0 ≤ σ < r, 1 ≤ j ≤ ℓ, and

C.J. Colbourn / Covering Arrays and Hash Families

a12 a22

··· ···

a1k a2k

··· ···

N rows

a11 a21 .. .

aN 2 b11 b21

··· ··· ···

aN k b11 b21

··· ··· ··· ···

M rows

aN 1 b11 b21 .. . bM 1

bM 1

···

bM 1

··· ···

105

a11 a21 .. .

a12 a22

··· ···

a1k a2k

aN 1 b1ℓ b2ℓ .. .

aN 2 b1ℓ b2ℓ

··· ··· ···

aN k b1ℓ b2ℓ

bM ℓ

bM ℓ

···

bM ℓ

Figure 3. The structure of A ⊗ B

1 ≤ j ′ ≤ ℓ′ , there exists a ρ with 1 ≤ ρ ≤ L so that the entry in cell (ρ, j) of B is σ and the entry in cell (ρ, j ′ ) of B′ is σ. Theorem 2.2 Suppose that there exists a CA(N ; 2, k, v), A, with profile (d1 , . . . , dk ), having r pure constant rows on symbols 0, . . . , r − 1. Further suppose that for each 1 ≤ i ≤ k and some 0 ≤ δi ≤ v − γ, there exists a CA(M + di + δi ; 2, ℓi , v), Bi , having γ −r +δi constant rows on symbols v −(γ −r)−δi , . . . , v −1 and possibly ⋆, which form the last (γ − r) + δi rows of Bi . Further suppose that for every 1 ≤ i1 < i2 ≤ k, Bi1 and k Bi2 are (M −(γ −r), r)-compatible. Then there exists a CA(N +M −γ; 2, i=1 ℓi , v).

Proof. Assume without loss of generality that the r pure constant rows of A form the last r rows, and remove them to form A′ with N − r rows. Form an array C with N + M − γ k rows and i=1 ℓi columns, indexing columns as (i, j) for 1 ≤ i ≤ k and 1 ≤ j ≤ ℓi . On the first N − r rows, column (i, j) is column i of A′ . For i = 1, . . . , k, define Ri = (ri,1 , . . . , ri,di ) to be the indices of the di rows in which A′ contains a ⋆ in column i. For 1 ≤ i ≤ k, 1 ≤ j ≤ ℓi , and 1 ≤ ρ ≤ M − (γ − r) place in row N − r + ρ and column (i, j) of C the entry in cell (ρ, j) of Bi . For 1 ≤ i ≤ k, 1 ≤ j ≤ ℓi , and 1 ≤ x ≤ di , in row rx and column (i, j) place the entry in cell (M − (γ − r) + x, j) of Bi . Consider columns (i1 , j1 ) and (i2 , j2 ) of the result. When i1 = i2 , all pairs of the form (σ, σ) are covered in the first N − r rows excluding those in Ri1 . Then in rows Ri1 and the last M − (γ − r) rows, all remaining pairs are covered because two different columns of Bi1 are selected (and δi ≤ v − γ). So suppose that i1 = i2 . The first N − r rows cover all pairs except possibly for (σ, σ) when 0 ≤ σ < r, which are covered by the remaining rows as a consequence of compatibility.  In applying Theorem 2.2 the main difficulty arises in ensuring compatibility among the {Bi }. The easiest application arises when the profile of A is (0, . . . , 0) and all of the {Bi } are identical: Corollary 2.3 If a CA(N ; 2, k, v) with r disjoint rows and a CA(M ; 2, ℓ, v) with s disjoint rows both exist, then a CA(N + M − min(v, r + s); 2, kℓ, v) exists having min(1, r + s − v) constant rows. Proof. By renaming symbols within each column and reordering rows, any CA(N ; 2, k, v) with r disjoint rows can be rewritten as a CA(N ; 2, k, v), A, in which the last r rows

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are constant on the symbols 0, . . . , r − 1. Then, whether or not A has any don’t care cells, it admits profile (0, . . . , 0) and r pure constant rows. By the same token, any CA(M ; 2, ℓ, v) with s disjoint rows can be rewritten as a CA(M ; 2, ℓ, v), B, in which the last s rows are constant on the symbols v − s, . . . , v − 1. Take di = δi = 0 and Bi = B for 1 ≤ i ≤ k, and γ = min(v, r + s). Every choice of two columns from the {Bi } either selects the same column of B twice, or selects two different columns of B. In both cases, only the first M − min(s, v − r) rows are considered, but pairs (σ, σ) for 1 ≤ σ ≤ r are covered within these rows. Apply Theorem 2.2. If r + s > v, the constant rows on symbols v − s, . . . , r − 1 in B yield constant rows in the result; when r + s ≤ v, choose any row and rename symbols to make it constant.  Compatibility among the {Bi } is vacuous when r = 0, and we obtain: Corollary 2.4 Suppose that A is a CA(N ; 2, k, v) having r′ pure constant rows and profile (d1 , . . . , dk ). Let 0 ≤ s ≤ v be an integer. Further suppose that for each 1 ≤ i ≤ k and some 0 ≤ δi ≤ v − s, there exists a CA(M + di + δi ; 2, ℓi , v), Bi , having s + δi constant rows on symbols v − s − δi , . . . , v − 1 and  possibly ⋆, which form the last s + δi k rows of Bi . Then there exists a CA(N + M − s; 2, i=1 ℓi , v) having r′ constant rows.

Proof. Apply Theorem 2.2 but take r = 0. Because A actually has r′ constant rows, each produces a constant row in the resulting array. 

A more interesting way to ensure compatibility is to require each of the {Bi } to contain constant rows other than the (v − r) + δi used in the construction. Indeed if Bi and Bj contain constant rows on 0, . . . , r − 1 among the first M − (γ − r) rows, they are (M − (γ − r), r)-compatible. Hence we obtain: Corollary 2.5 Suppose that there exists a CA(N ; 2, k, v), A, with profile (d1 , . . . , dk ), having r′ pure constant rows, and let 0 ≤ r ≤ r′ be an integer. Let γ ≥ r be an integer. Further suppose that for each 1 ≤ i ≤ k and some 0 ≤ δi ≤ v − γ, there exists a CA(M + di + δi ; 2, ℓi , v), Bi , having γ + δi constant rows. Then there exists a k CA(N + M − γ; 2, i=1 ℓi , v) having r′ constant rows.

Proof. Rename symbols in A so that the last r rows are constant on 0, . . . , r − 1 and the first r′ − r are constant on r, . . . , r′ − 1. Rename symbols and reorder rows in each of the {Bi } so that the first r rows are constant on symbols 0, . . . r − 1 and possibly ⋆; and the last γ − r + δi are constant on symbols v − (γ − r) − δi , . . . , v − 1 and possibly ⋆. Apply Theorem 2.2. The r constant rows on symbols 0, . . . r − 1 in each of the {Bi }, and the r′ − r constant rows on symbols r, . . . , r′ − 1 in A, produce constant rows in the resulting array. 

Now we consider some applications. When q is a prime power there is a CA(q 2 ; 2, q+ 1, q), which necessarily has only one constant row, and no ⋆ entries. Using this to construct A and each of the {Bi }, Corollary 2.3 produces a CA(2q 2 − 2; 2, (q + 1)2 , q). In order to illustrate Corollary 2.4, we produce some profiles for a CA(42; 2, 8, 6) having r pure constant rows. (The array itself is from [107]; we simply analyze its properties here.) When p1 is a profile for r pure constant rows, p2 ≤ p1 , and r′ ≤ r, p2 is a profile for r′ pure constant rows. We also show the number of constant rows s in various small CA(M ; 2, ℓ, 6)s from [39,45,107,129].

C.J. Colbourn / Covering Arrays and Hash Families

r 6 5 4 3 2 0

profile 04 14 06 11 21 05 11 22 , 07 31 03 15 04 13 21 02 16 , 03 14 21 , 04 12 22 , 05 23 , 05 11 21 31 , 06 32

M 36 42 54 60 65 69

ℓ 3 8 12 16 20 30

s 6 6 3 2 2 6

M 37 46 56 62 66 70

ℓ 4 9 13 17 21 32

s 5 1 2 3 6 6

M 39 49 58 63 67 71

107

ℓ 5 10 14 18 23 35

s 4 3 3 2 6 2

M 41 52 59 64 68 72

ℓ 6 11 15 19 27 36

s 5 3 3 2 6 1

Applying Corollary 2.4 with s = 6, M ∈ {66, 67, 68, 69}, and various values of r′ , we obtain the following arrays with M + 42 − 6 rows, each having at least r constant rows: r↓ 6 4 3 0

CA(102; 2, K, 6) K profile 176 04 14 182 05 11 22 182 05 11 22 186 05 23

CA(103; 2, K, 6) K profile 200 04 14 202 05 11 22 204 03 15 208 02 16

CA(104; 2, K, 6) K profile 228 04 14 229 05 11 22 231 03 15 234 02 16

CA(105; 2, K, 6) K profile 248 04 14 248 04 14 250 03 15 252 02 16

Naturally in the cases when r = 0 there is nonetheless at least one disjoint row. Effective applications of Corollary 2.4 seem to necessitate a fairly detailed analysis of the array A to determine numbers of disjoint rows and corresponding profiles. Now we turn to the most surprising application, that of Corollary 2.5. Lemma 2.6 Suppose that there exists a CA(N ′ ; 2, k, v) with v 2 ≤ N ′ < v(v + 1).

1. If a CA(M ′ ; 2, ℓ, v − 1) having s constant rows also exists, there exists a CA((M ′ − s) + (N ′ − v); 2, (k − 1)ℓ, v − 1) having v − 1 constant rows. 2. If a CA(M ′ ; 2, ℓ, v − 1) having s constant rows also exists and M ′ ≥ v(v − 1), there exists a CA((M ′ − s) + (N ′ − v); 2, (k − 1)ℓ + 2, v − 1) having v − 1 constant rows. 3. If a CA(M ′ ; 2, ℓ1 , v − 1) and a CA(M ′ − (v − 1); 2, ℓ2 , v) both having s constant rows also exist, there exists a CA((M ′ − s) + (N ′ − v); 2, (k − 1)ℓ1 + ℓ2 , v − 1) having v − 1 constant rows.

Proof. Taking ℓ2 = 0 and ℓ1 = ℓ in the third statement implies the first. Taking ℓ2 = 2 and ℓ1 = ℓ in the third statement implies the second, because a CA(M ′ − (v − 1); 2, 2, v) exists with v − 1 disjoint rows when M ′ − (v − 1) ≥ (v − 1)2 . So we establish the third statement. Form a CA(N ′ ; 2, k, v). In each column there is a symbol that occurs only v times, as N ′ < v(v + 1). Rename symbols so that there is a constant row of symbol v − 1, and symbol v − 1 occurs exactly v times in the last column. Delete this constant row and change all occurrences of symbol v − 1 to ⋆ to form a CA(N ′ − 1; 2, k, v − 1) having profile (v − 1)k . (It may have more ⋆ entries in columns other than the last, but its profile dominates (v − 1)k .) Replace the v − 1 ⋆ entries in the final column by entries 0, . . . , v − 2, using each symbol exactly once. Then the v − 1 corresponding rows are pairwise disjoint, and the result is a CA(N ′ − 1; 2, k, v − 1) having v − 1 pure constant rows, and profile (v − 1)k−1 01 . Apply Corollary 2.5 with γ = r = s, r′ = v − 1, N = N ′ − 1, and M = M ′ − (v − 1). 

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2.2. Roux-type Products The most effective applications of direct products for strength two employ the presence of constant rows. Unfortunately, in some cases we must sacrifice columns to obtain many constant rows. Let us consider a motivating example. When q is a prime power, the standard CA(q 2 ; 2, q, q) from the finite field on q elements has q disjoint constant rows, but its extension to a CA(q 2 ; 2, q + 1, q) can have at most one, no matter how the symbols are relabeled. Applying Theorem 2.2 to two CA(25; 2, 6, 5)s produces a CA(48; 2, 36, 5). Instead using a CA(25; 2, 6, 5) and a CA(25; 2, 5, 5) with five disjoint constant rows yields a CA(45; 2, 30, 5); three fewer rows suffice, but six columns have been lost in the result. In [56] a generalization is treated that that enables us to obtain a CA(45; 2, 35, 5), sacrificing one column in the product rather than one column in an ingredient.

A1

A2

D

X

Figure 4. A partitioned covering array (PCA)

We consider covering arrays exhibiting a specific structure. Consider a CA(N ; 2, k1 + k2 , v), shown in Figure 4. Here A1 , A2 , and X are (N − v) × k1 , (N − v) × k2 , and v × k2 arrays, respectively. However D is a v × k1 array with a specific structure, namely that every column is a permutation of {1, . . . , v}. When a CA(N ; 2, k1 + k2 , v) admits such a partition, it is a partitioned covering array PCA(N ; 2, (k1 , k2 ), v). The structure is not altered by applying (possibly different) permutations to the v symbols in each column, and hence without loss of generality D can be assumed to be the matrix P in which each column is the identity permutation. When q is a prime power, an OA(2, q + 1, q) yields an PCA(q 2 ; 2, (q, 1), q). Now we turn to the main product construction for covering arrays: Theorem 2.7 If a PCA(N ; 2, (k1 , k2 ), v) and a PCA(M ; 2, (ℓ1 , ℓ2 ); v) both exist, then a PCA(N + M − v; 2, (k1 ℓ1 , k1 ℓ2 + k2 ℓ1 ), v) also exists. Proof. Take a PCA(N ; 2, (k1 , k2 ), v) with a partition as in Figure 4 into A1 , A2 , D and X; and an PCA(M ; 2, (ℓ1 , ℓ2 ), v) with partition B1 , B2 , E, and Y. We suppose without loss of generality that D and E consist of column identity permutations, and we write each as P. We further suppose that each of the columns of X and Y has the property that the i + 1st entry does not exceed i. Form an array as in Figure 5. In the products of the form Ai ⊗ Bj , the first N − v rows arise from Ai while the next M − v arise from Bj , as shown in Figure 3. Here ℓ1 X is obtained by repeating the array X ℓ1 times and k1 (Y) is obtained by repeated each column of Y k1 times. P is a matrix of identity permutations of appropriate dimension. We claim that the result R is an PCA(N +M −v; 2, ((k1 ℓ1 , k1 ℓ2 +k2 ℓ1 ), v). Consider two columns c1 , c2 of the result. Suppose that column ci corresponds to column αi of A and column βi of B. We tabulate cases (indicating cases that cannot arise by ♣), taking symmetry between A and B and between c1 and c2 into account:

C.J. Colbourn / Covering Arrays and Hash Families

109

A1 ⊗ B1

A2 ⊗ B1

A1 ⊗ B2

P

ℓ1 X

k1 (Y)

Figure 5. The product of two PCAs

k1 ≤ α2 = α1

1 3 4 4 6 7

α2 = α1

♣ 1 2 2 ♣ 5

♣ 6 ♣ ♣ ♣ ♣

5 7 ♣ ♣ ♣ ♣

α2 ≤ k1

β1 > ℓ1

β2 = β1 β1 =  β2 ≤ ℓ1 β2 > ℓ1 β2 ≤ ℓ1 β2 = β1 ℓ1 < β1 = β2

α1 > k1

α2 > k1

β1 ≤ ℓ1

α1 = α2 ≤ k1

B↓

α1 ≤ k1 α2 = α1

A→

2 4 ♣ 8 ♣ ♣

2 4 8 ♣ ♣ ♣

We treat each case. In Cases 1, 3, 4, 6, and 7 when α1 = α2 and α1 , α2 ≤ k1 , the first N − v rows cover all pairs except possibly {(i, i) : 0 ≤ i < v}. In each of these cases, because c1 and c2 select different columns of D, all remaining pairs are covered. It remains to treat cases 2, 5, and 8. For cases 2 and 5, α1 = α2 and β1 = β2 ≤ ℓ1 . Then in the first N − v and last v rows, α1 and α2 select different columns of A, and hence all pairs are covered. Finally in case 8, α2 ≤ k1 < α1 and β1 ≤ ℓ1 < β2 . Let column α1 of X be (x0 , . . . , xv−1 )T and column β2 of Y be (y0 , . . . , yv−1 )T . Then in the first N − v rows all pairs are covered except {(xi , i) : 0 ≤ i < v}, and in the next M − v all are covered except {(i, yi ) : 0 ≤ i < v}. It follows that prior to the last v rows, a pair can be uncovered only if (xi , i) = (j, yj ) for some 0 ≤ i, j < v. Then because xi ≤ i and yi ≤ i for 0 ≤ i < v, we have xi ≤ i = yj ≤ j = xi , so all are equal, and the pair is (i, i) = (xi , yi ); this pair is covered in the last v rows. Then R is a PCA(N + M − v; 2, (k1 ℓ1 , k1 ℓ2 + k2 ℓ1 ), v) because R has a v × k1 ℓ1 subarray consisting of (column) identity permutations in the last v rows.  A substantial improvement is still possible. One can, on occasion, find a larger submatrix in the result R that contains column identity permutations, and hence provide a better ingredient for the next iteration of the recursion. If the second PCA were to contain within B1 an v × η subarray in which every column is a permutation of {0, . . . , v − 1}, then let us examine the impact on the covering array R constructed in Theorem 2.7. Each column of B1 is replicated k1 + k2 times in total, and hence R contains a v × η(k1 + k2 ) subarray in which every column is a permutation. If η(k1 + k2 ) > k1 ℓ1 , we can permute symbols within each column, and permute rows: Theorem 2.8 When B1 in Theorem 2.7 contains a v × η subarray whose columns are permutations, the result is a PCA(N +M −v; 2, (η(k1 +k2 ), (ℓ1 −η)(k1 +k2 )+k1 ℓ2 ), v).

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For example, the OA(q 2 ; 2, q + 1, q) for q a prime power contains a second q × q subarray of column permutations. As stated here, Theorem 2.7 is a small generalization of the result in [56]; unlike Theorem 2.2, however, it does not exploit don’t-care positions. In order to do so, we generalize here to use more than two ingredient arrays. Let A be a PCA(N ; 2, (k1 , k2 ), v) with k = k1 + k2 and partition A1 , A2 , D, and X. Then A has restricted profile (d1 , . . . , dk ) if, for 1 ≤ i ≤ k, there are at least di ⋆ entries in column i of (A1 A2 ). We also require a definition from [56]. An SCA(N ; 2, (k1 , k2 ), v) is a PCA(N ; 2, (k1 , k2 ), v) with partition A1 , A2 , P, and Z in which Z is the all-zero matrix and P is a matrix of column identity permutations of appropriate dimension. Now we adjust the notion of compatibility to suit our purposes. Let B be an N × k matrix on v symbols, and B′ be an N ′ × k ′ matrix on v symbols. Then B is (M, c)-equalcompatible with B′ if M ≤ min(N, N ′ ), and for every 1 ≤ σ < v, each of columns c + 1, . . . , k of B, and every column of B′ , there exists a ρ with 0 ≤ ρ < M so that the entry in row N − ρ in the column of B is σ and the entry in row N ′ − ρ in the column of B′ is σ. Further, B is (M, c)-down-compatible with B′ if M ≤ min(N, N ′ ), and for every 1 ≤ σ < v, each of columns c + 1, . . . , k of B and every column of B′ , there exists a ρ with 0 ≤ ρ < M so that the entry in row N − ρ in the column of B is σ and the entry in row N ′ − ρ in the column of B′ is 0. Theorem 2.9 Suppose that there exist 1. an SCA(N + v; 2, (k1 , k2 ), v), A, with k = k1 + k2 and restricted profile (d1 , . . . , dk ) and partition A1 , A2 , P, and Z; 2. for 1 ≤ i ≤ k1 , a PCA(M + v + di ; 2, (ℓi,1 , ℓi,2 ); v), Bi , with ℓi = ℓi,1 + ℓi,2 and partition Bi,1 , Bi,2 , P, and Yi in which every column (y1 , . . . , yv )T of Yi has yi+1 ≤ i for 0 ≤ i < v; and 3. for k1 < i ≤ k, an SCA(M + v + di ; 2, (ℓi , 0), v), Bi with partition Bi,1 , O, P, and O (equivalently, a CA(M + di ; 2, ℓi , v) having v constant rows). Suppose that for 1 ≤ i = i′ ≤ k1 , (Bi,1 Bi,2 ) is (M, ℓi,1 )-equal-compatible with (Bi′ ,1 Bi′ ,2 ), and that for 1 ≤ i ≤ k1 and k1 < i′ ≤ k, Bi is (M, ℓi,1 )-down-compatible k k1 k1 ℓi,2 + i=k1 +1 ℓi ), v) also ℓi,1 , i=1 with Bi′ . Then an SCA(N + M + v; 2, ( i=1 exists. Proof. Index the columns of the result R by {(i, j) : 1 ≤ i ≤ k, 1 ≤ j ≤ ℓi }. We describe how to form column (i, j) of R. Let C1 be the ith column of (A1 A2 ) and let C2 be the jth column of (Bi,1 Bi,2 ) when i ≤ k1 , or the jth column of Bi,1 when i > k1 . Let C3 be a column of P when i ≤ k1 and j ≤ ℓi,1 , the jth column of Yi when i ≤ k1 and j > ℓi,1 , and a column of Z otherwise. Replace the ⋆ entries in C1 by the first di entries of C2 to form C′1 , and let C′2 be the last M entries of C2 . Form the column with N + M + v entries by vertically juxtaposing C′1 , C′2 , and C3 . Then R has the required number of rows, columns, and symbols, and admits the partition specified. It remains to verify that it is a covering array. Consider columns indexed by (i, j) and (i′ , j ′ ). First consider the cases when i = i′ . When i, i′ > k1 ; i, i′ ≤ k1 , j ≤ ℓi,1 , and ′ j ≤ ℓi′ ,1 ; or i ≤ k1 < i′ and j ≤ ℓi,1 , all pairs are covered in the first N rows and last v rows, because the restriction to these rows gives two different columns of A. For cases with i, i′ ≤ k1 , all pairs with two unequal symbols are covered in the first N rows; the

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pair (0,0) is covered in the last v rows, and the other pairs with equal symbols are covered in the remaining rows because Bi is (M, ℓi,1 )-equal-compatible with Bi′ . It remains to treat cases with i ≤ k1 < i′ and j > ℓi,1 . In the first N rows, all pairs but possibly {(σ, 0) : 0 ≤ i < v} are covered, and (0, 0) is covered in the last v rows. The remainder are covered in the rows arising from Bi and Bi′ because Bi is (M, ℓi,1 )-down-compatible with Bi′ . It remains to treat columns indexed by (i, j) and (i, j ′ ). When i ≤ k1 , the two columns contain two different columns of Bi and hence all pairs are covered. When i > k1 , within the rows arising from Bi all pairs of unequal symbols are covered; because the column from A is chosen twice, all pairs with equal symbols are covered in the first N − v rows.  Not surprisingly, the technical requirements of Theorem 2.9 make it hard to apply. Theorem 2.7 applied to a PCA and an SCA is a consequence of Theorem 2.9 using restricted profile (0, 0, . . . , 0) and forming each of the {Bi } from a single PCA. Instead taking k1 = 0 and k = k2 , no equal- or down-compatibility needs to be checked, and Theorem 2.9 yields Corollary 2.4 with s = v, a ‘direct product construction’. Lemma 2.6 essentially shows that a CA(N + v + 1; 2, k, v + 1) with v(v + 1) ≤ N < (v + 1)2 yields an SCA(N + v; 2, k, v) having a restricted profile that dominates v k−1 01 . To exploit this, we first need a PCA(M + v; 2, (ℓi,1 , ℓi,2 ), v) Bi with partition Bi,1 , Bi,2 , P, and Y. To apply Theorem 2.9 effectively, we further require that Bi,1 and Bi,2 be (M − v, ℓi,1 )-equal-compatible. We see no easy way to ensure this in general, so we adopt a simple trick when ℓi,2 = 1. Adjoin a constant row containing symbol v − 1 to (Bi,1 Bi,2 ) to form (Bi,1 Bi,2 ) having M + v + 1 rows, and ensure that the first v rows are the rows containing v − 1 in the column of B′i,2 , other than the constant row of symbol v − 1. Then B′i,1 is (M + 1 − v, ℓi,1 )-equal-compatible with B′i,2 . Lemma 2.10 Suppose that a CA(N + v + 1; 2, k, v + 1) exists with v(v + 1) ≤ N < (v + 1)2 ; that a PCA(M + v; 2, (ℓ, 1), v) exists; and that κ = 1 or a PCA(v 2 ; 2, κ + 1, v) exists. Then there exists a PCA(N + M + 1; 2, ((k − 1)ℓ, k − 1 + κ), v). Proof. This is a variant of Theorem 2.9 using the SCA(N + v; 2, k, v) with restricted profile v k−1 01 as A. To form Bi with 1 ≤ i < k, order the rows of the PCA(M + v; 2, (ℓ, 1), v) so that the first v rows contain v−1 in the last column, and insert a constant row containing v − 1 immediately thereafter to form a PCA(M + 1 + v; 2, (ℓ, 1), v). To form Bk , for each 0 ≤ σ < v let Rσ consist of all positive values ρ − v for which Bi contains σ in row ρ in the last column. When κ = 1, for each 0 ≤ σ < v, ensure that σ appears as a row of Bk among the rows indexed by Rσ . When κ > 1, partition the rows of the CA(v 2 ; 2, κ + 1, v) for Bk into v classes C0 , . . . , Cv−1 placing a row in class Cσ if it has the symbol σ in column κ + 1. Then delete column κ + 1, and rename symbols so that C0 consists of v constant rows. Then order the rows of Bk , possibly together with rows consisting entirely of ⋆, so that for 0 ≤ σ ≤ v − 2, the rows of Cσ appear in the rows indexed by Rσ . Place a constant row of symbol v − 1 as the first row of Bk (thereby aligning it to form a constant row of v − 1 throughout). Apply Theorem 2.9.  This can improve upon Lemma 2.6. For example, using a CA(110+10+1; 2, 12, 11) and a PCA(92+10;2,(5,1),10) (the latter from an incomplete transversal design, see [23,50]), Lemma 2.10 gives a PCA(203;2,(55,12),10), while Lemma 2.6 gives a

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CA(205;2,66,10). In this case, M < v 2 so κ = 1. Similarly, using a CA(110 + 10 + 1; 2, 12, 11), a PCA(106+10;2,(10,1),11) from [53], and a CA(100;2,4,10), Lemma 2.10 gives a PCA(217;2,(110,14),10). We expect that with patience one can find a number of such applications. Nevertheless, the accumulation of conditions that must be met seems to have led to the land of diminishing returns; one needs a construction with the power of Theorem 2.9 for which ingredients are more easily found.

3. Perfect Hash Families Theorem 2.1 constructs a CA(N +M ; 2, kℓ, v) from a CA(N ; 2, k, v) and a CA(M ; 2, ℓ, v). In Section 2.1, generalizations to use constant rows and don’t care positions have been treated; then in Section 2.2 these were generalized further as ‘cut-and-paste’ or Rouxtype constructions. The further generalization of Roux-type constructions to higher strengths has been alluded to, and we discuss it no further here. Instead we return to the basic direct product, in order to take a different view. When A is a CA(N ; 2, k, v) and B is a CA(M ; 2, ℓ, v), a column of the result C is obtained by selecting a column of A and a column of B and vertically juxtaposing them. In essence, then, the construction specifies which columns of A and B to juxtapose in order to form the column of C. Suppose that we form a 2 × kℓ array D = (dij ) so that, for 1 ≤ i ≤ k and 1 ≤ j ≤ ℓ, d1,(i−1)ℓ+j = i and d2,(i−1)ℓ+j = j. Then the array C from Theorem 2.1 is formed by replacing each symbol σ in the first row of D by the column indexed by σ in A, and replacing each symbol τ in the second row of D by the column indexed by τ in B. That the result C is a CA(M ; 2, ℓ, v) follows immediately from the fact that for any two columns in D there is a row in which they contain different symbols. Our objective is to generalize to higher strengths by considering the properties of the ‘pattern’ array D used to select columns from the ingredient covering array(s). The key property is that, in this pattern array, every t columns select t different columns of the ingredient arrays – and we encounter a well-studied combinatorial object. A perfect hash family PHF(N ; k, w, t) is an N × k array on w symbols, in which in every N × t subarray, at least one row consists of distinct symbols. The smallest N for which a PHF(N ; k, v, t) exists is the perfect hash family number, denoted PHFN(k, v, t). Figure 6 shows a PHF(6; 12, 3, 3). For instance, in columns 1, 8, and 9, the fourth row contains 2 0 1, and it is the only row in which we find three distinct symbols in these columns. ⎡

⎤ 012212201100 ⎢0 2 1 0 2 2 2 1 0 1 2 1⎥ ⎢ ⎥ ⎢1 0 0 2 2 2 1 1 2 1 0 2⎥ ⎢ ⎥ ⎢2 0 1 1 2 0 2 0 1 1 2 1⎥ ⎢ ⎥ ⎣2 0 2 1 2 1 0 2 2 1 1 0⎦ 201211220121 Figure 6. A PHF(6; 12, 3, 3)

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Mehlhorn [103] introduced perfect hash families as an efficient tool for compact storage and fast retrieval of frequently used information. In this setting, each row defines a hash function from a domain of size k to a range of size v; we employ the array formulation instead. Stinson, Trung, and Wei [124] establish that perfect hash families can be used to construct separating systems, key distribution patterns, group testing algorithms, cover-free families, and secure frameproof codes; see also [21,125]. Perfect hash families have also been applied in broadcast encryption [33,65] and threshold cryptography [18]. Finally, perfect hash families arise as ingredients in some recursive constructions for covering arrays; we examine this in detail here. An older survey on PHFs is given in [61]; for recent results on the existence of perfect hash families see [55,99,131] and references therein. Our reason for interest in them here follows: Theorem 3.1 [15,98] If a PHF(s; k, m, t) and a CA(N ; t, m, v) both exist then a CA(sN ; t, k, v) exists. Proof. Let B = (bij ) be an s × k array on m symbols forming a PHF(s; k, m, t). Let A = (aij ) be an N × m array on v symbols forming a CA(N ; t, m, v). We produce an sN × k array C = (cij ) as follows. For each 1 ≤ i ≤ s, 1 ≤ j ≤ N , and 1 ≤ ℓ ≤ k, set c(i−1)N +j,ℓ = aj,bi,ℓ . The verification that C is a CA(sN ; t, k, v) is straightforward; one needs only check that every t-way interaction is covered. Consider the t-way interaction {(γ1 , ν1 ), . . . , (γt , νt )}. Because B is a perfect hash family of strength t, it is also a perfect hash family of strength t′ for all t′ ≤ t. Therefore there is a row ρ of B in which bρ,γi = bρ,γj whenever νi = νj (and hence γi = γj ). Set di = bρ,γi for 1 ≤ i ≤ t. In columns (d1 , . . . , dt ) of A, there is a row τ in which aτ,di = νi , because A is a covering array of strength t and di = dj when νi = νj . But then c(ρ−1)N +τ,γi = νi for 1 ≤ i ≤ t, and the t-way interaction is covered in C.  Less formally, the perfect hash family B is used as a ‘pattern’ to select columns from the covering array A, so that every symbol σ of B is replaced by the entire column of A that is indexed by σ. An immediate improvement arises by taking standardizing the covering arrays: Corollary 3.2 If a PHF(s; k, m, t) and a CA(N ; t, m, v) both exist then a CA(1+s(N − 1); t, k, v) exists. Proof. When the covering array is standardized, the constant row is repeated s times, and s − 1 copies of it can be removed.  We examine generalizations of this column replacement technique, by varying the properties of the hash family and covering array involved, and by permitting the use of many covering arrays rather than a single one. First we are concerned with the construction of perfect hash families. 3.1. Perfect Hash Families, Packings, and Codes A packing array ΠA(N ; t, k, v) is an N × k array with symbols from a v-set so that in every t × k subarray, no two columns are equal. Equivalently, no two columns of the N ×k array agree in t or more cells; hence the term ‘packing’ array. These were explicitly studied in the case when t = 2 as transversal packings [122,123], although they have

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been long explored as v-ary error-correcting codes. We first provide a few definitions. Let x = (x1 , x2 , . . . , xN ) and y = (y1 , y2 , . . . , yN ) be v-ary vectors of length N . The Hamming distance between x and y is d(x, y) := |{i|xi = yi }|. An (N, K, D, v)-code is a set C of K vectors (codewords) in {1, . . . , q}N such that the Hamming distance between any two distinct vectors in C is at least D. Codes over an alphabet of size v are v-ary codes. (See [94], for example.) Forming an N × K array from an (N, K, D, v)code by taking codewords as columns yields a ΠA(N ; N −D, K, v). Indeed the converse holds as well. While codes are much better studied than packing arrays, we adopt the array vernacular here. We mention two well known constructions used in [46]: Lemma 3.3 A set of N − 2 mutually orthogonal latin squares of order v is equivalent to a ΠA(N ; 2, v 2 , v). The OA(q t ; t, q + 1, q) from polynomials over the finite field establishes a result essentially due to Bush [27]: Lemma 3.4 When q is a prime power and 1 ≤ t ≤ q, there is a ΠA(q + 1; t, q t , q). s s Lemma 3.5 [1]  An OA(n ; s, k, n) – transposed – yields a PHF(k; n , n, t) whenever t k > (s − 1) 2 .

More generally, we have:

 Lemma 3.6 When N > (s − 1) 2t , a ΠA(N ; s, k, w) is a PHF(N ; k, w, t).

Proof. Let A be a ΠA(N ; s, k, w). Choose t columns of A, and consider the N × t  subarray induced on them. There are 2t pairs of columns, and for each pair there are at  most s − 1 entries in which they agree. Hence when N > (s − 1) 2t there is at least one row in which no two of the selected columns contain the same entry.  In addition to Lemmas 3.5 and 3.6, better results are often available. Theorem 3.7 [19] Let s ≥ 2 and t ≥ 2. When q is a sufficiently large prime power, there is a PHF(s(t − 1); q s , q, t). Blackburn and Wild [19] also prove that s(t − 1) is a lower bound on the number of rows in a PHF arising from a linear OA, and hence the PHF produced in these cases is an optimal linear PHF. Some explicit computations have been undertaken to determine those prime powers for which such an optimal linear PHF can be constructed: Theorem 3.8 [11,12,13,17] 1. An optimal linear PHF(6; q 2 , q, 4) exists if and only if q ≥ 11 is a prime power and q = 13. 2. An optimal linear PHF(6; q 3 , q, 3) exists if and only if q ≥ 11 is a prime power. Many results are known for numbers of rows intermediate between that prescribed by Lemma 3.6 and Theorem 3.7; each uses linear orthogonal arrays [54]. We repeat some of them that are useful in making covering arrays.

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Theorem 3.9 [54] Let p be a prime. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

A PHF(9; p4 , p, 3) exists when p ≥ 17. A PHF(8; p4 , p, 3) exists when p ≥ 19. A PHF(12; p3 , p, 4) exists when p ≥ 17. A PHF(11; p3 , p, 4) exists when p ≥ 29. A PHF(10; p3 , p, 4) exists when p ≥ 251 and p ∈ {257, 263}. A PHF(10; p2 , p, 5) exists when p ≥ 19. A PHF(9; p2 , p, 5) or a DHF(9; p2 , p, 5, 4) exists when p ≥ 41. A PHF(8; p2 , p, 5) exists when p ≥ 241 and p ∈ {251, 257}. A PHF(15; p2 , p, 6) exists when p ≥ 29. A PHF(14; p2 , p, 6) or a DHF(14; p2 , p, 6, 5) exists when p ≥ 41. A PHF(13; p2 , p, 6) exists when p ≥ 73.

Some extensions to orders that are not powers of primes appear in [63]. Here we mention two recursive constructions for perfect hash families, and explore the topic further in Section 5. Blackburn [16] gives a simple product construction, composition, which is itself a column replacement technique: Theorem 3.10 Suppose there exist a PHF(N0 ; k, x, t) and a PHF(N1 ; x, v, t). Then there exists a PHF(N0 N1 ; k, v, t). Atici et al. [7] give a product type construction: Theorem 3.11 Suppose that a PHF(N1 ; k0 k1 , v, t), a PHF(N2 ; k2 , k1 , t − 1), and a PHF(N3 ; k2 , v, t) all exist. Then there exists a PHF(N1 N2 + N3 ; k0 k2 , v, t). Stinson, Wei, and Zhu [127] develop further constructions.

4. Related Hash Families By weakening the requirements on the sets of columns to be separated, a number of variants of PHFs have been explored. 4.1. Distributing hash families In [46], a generalization of perfect hash families is examined in order to construct covering arrays. An N × t array A on w symbols (with columns C = {1, . . . , t}) is (t, v)distributing if, for every partition {C1 , . . . , Cv } of C into v parts, there is at least one row of A, (a1 , . . . , at ), in which ai = aj only if i and j belong to the same class of the partition. An N × k array is (t, v)-distributing if every N × t subarray is (t, v)-distributing; such an array is a distributing hash family, and is denoted by DHF(N ; k, w, t, v). An (N ; k, v, {w1 , w2 , . . . , wt })-separating hash family is an (N ; k, v)-hash family H that satisfies the property: For any C1 , C2 , . . . , Ct ⊆ {1, 2, . . . , k} such that |C1 | = w1 , |C2 | = w2 , . . . , |Ct | = wt , and Ci ∩ Cj = ∅ for every i = j, there exists at least one function h ∈ H such that {f (y) : y ∈ Ci } ∩ {f (y) : y ∈ Cj } = ∅. The notation SHF(N ; k, v, {w1 , w2 , . . . , wt }) is used. See, for example, [2,113,119]; and see [10] for the closely related notion of ‘partially hashing’.

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Observation 4.1 Let A be an N × k array. The following are equivalent:

1. A is a DHF(N ; k, w, t, v). 2.  A is a SHF(N ; k, w, {w1 , w2 , . . . , wv }) for every set {w1 , w2 , . . . , wv } with v i=1 wi = t and wi ≥ 0 for 1 ≤ i ≤ v.

Observation 4.2 Let A be an N × k array. The following are equivalent:

1. A is a PHF(N ; k, v, t). 2. A is an SHF(N ; k, v, {w1 , w2 , . . . , wt }) with w1 = · · · = wt = 1. 3. A is a DHF(N ; k, w, t, v) with v ≥ t.

By Observation 4.1, an SHF(N ; k, w, 1t−1 21 ) is the same as a DHF(N ; k, w, t+1, t) because the only partition of t + 1 elements into t nonempty parts has one part with two elements and the remainder with one. However, this equivalence does not extend further. While an SHF requires separation of a particular type of partition, a DHF can require the separation of many types of partitions. Stinson, Wei, and Chen [126] give an SHF(3;4,3,{2, 2}) that fails to separate partitions of type {1, 3} (and hence is not a DHF(3;4,3,4,2)); they also give an SHF(2;4,3,{1, 3}) that fails to separate partitions of type {2, 2} (and hence is not a DHF(2;4,3,4,2)). Distributing hash families weaken the conditions on perfect hash families, while strengthening the conditions on certain separating hash families, by requiring (in general) that more than one but fewer than all partitions of t columns be separated. Distributing hash families often require fewer rows than perfect hash families of the same strength. We collect some further easy observations. Observation 4.3 1. When s > 1, a DHF(N ; k, v, t, s) is also a DHF(N ; k, v, t, s − 1). 2. A DHF(N ; k, v, t, s) yields a DHF(N ; k + 1, v + 1, t, s). 3. When t > 0, a DHF(N ; k, v, t, s) is a DHF(N ; k, v, t − 1, min(s, t − 1)). Lemma 3.6 points to a useful generalization. We need a preliminary definition. The Turán number T (t, v) is the largest number of edges in a graph on t vertices that contains no complete subgraph of size v + 1. Turán [130] determined T (t, v) exactly, as follows. Write a = ⌊t/v⌋, and form a complete multipartite graph M with v classes, of which t − av have size a + 1 and (a + 1)v − t have size a. Then T (t, v) is the number of edges in M . Lemma 4.4 When N > (s − 1)T (t, v), a ΠA(N ; s, k, w) is a DHF(N ; k, w, t, v). Proof. Let A be a ΠA(N ; s, k, w). Choose t columns of A, and consider the N × t subarray induced on them. Consider a partition of the indices of the t columns into v classes; this defines a graph G on t vertices containing no complete subgraph of size v +1 and hence G has T (t, v) or fewer edges. Each edge of G indicates a pair of columns that are to contain distinct entries; as before, for each pair there are at most s − 1 entries in which they agree. Hence when N > (s − 1)T (t, v) there is at least one row in which no two of the selected columns joined by an edge of G contain the same entry.  For certain parameters, substantial improvements on Lemmas 3.5 and 4.4 are available.

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Theorem 4.5 [54] Let p be a prime. 1. A DHF(10; p3 , p, 4, 3) exists when p ≥ 31 and p ∈ {37, 41}. 2. A DHF(8; p2 , p, 5, 3) exists when p ≥ 61 and p ∈ {67, 71, 79, 83, 89, 103, 113, 137, 139}. 3. A DHF(13; p2 , p, 6, 4) exists when p ≥ 67. 4.2. Partitioning hash families An N × t array A on v symbols (with columns C = {1, . . . , t}) is (t, s)-partitioning if, for every partition {C1 , . . . , Cs } of C into at least two and at most s nonempty parts, there is at least one row of A, (a1 , . . . , at ), in which ai = aj if and only if i and j belong to the same class of the partition. An N × k array is (t, s)-partitioning if every N × t subarray is (t, s)-partitioning; such an array is a partitioning hash family, denoted by PaHF(N ; k, v, t, s). Sloane [118] explores the use of intersecting codes. An intersecting code is a linear code over Fq in which, for every two nonzero codewords, there is at least one coordinate in which the two codewords are both nonzero. In [118, Theorem 3] it is shown that for binary intersecting codes, choosing any three codewords a, b, c, there is a coordinate position in which a and b agree and c differs (and hence another in which a and c agree but b differs, and a third in which b and c agree but a differs). When the binary code has dimension d (2d codewords) and length M , this gives a DHF(M ; 2d , 2, 3, 2). The stronger requirement holds more generally. Lemma 4.6 [46] Every DHF(M ; k, 2, t, 2) has the property that for every t columns C = {c1 , . . . , ct } and every partition of C into two nonempty classes C1 and C2 , there is a row in which the entries in columns in C1 are the same, the entries in columns in C2 are the same, but the entries in columns in C1 differ from those in columns of C2 . Equivalently, it is a PaHF(M ; k, 2, t, 2). Proof. By the definition of a DHF, there is a row in which the entries in columns in C1 all differ from entries in columns in C2 . Because the array is binary, all entries in columns in C1 must agree in this row; similarly, all entries in columns in C2 must agree in this row.  A PaHF(N ; k, t, t+1, t) is an SHF(N ; k, t, {w1 , . . . , wt }) with w1 = · · · = wt−1 = 1 and wt = 2. This is an example of a strong separating hash family [113]. However, even in this restricted case the SHF need not be a PaHF, because a partition of t + 1 into fewer than t classes need not contain a row with the same symbol in the columns of each class but different symbols in the columns of different classes. Indeed, partitioning hash families appear to be challenging to construct! 4.3. Heterogeneous hash families Colbourn and Torres-Jiménez [59] relax the requirement that each hash function (row) have a range of the same size (the same number of symbols, respectively). A heteroge-

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neous hash family, denoted HHF(N ; k, (v1 , . . . , vN )), is an N × k array in which the ith row contains (at most) vi symbols for 1 ≤ i ≤ N .  Often we write (v1 , . . . , vN ) in expoc nential notation: v1u1 · · · vcuc means that the N = i=1 ui rows can be partitioned into classes, so that in the ith class there are ui rows each employing (at most) vi symbols. The definitions for PHF, DHF, and SHF extend naturally to perfect, distributing, and separating heterogeneous hash families; we extend the notation as follows: PHF(N ; k, w, t) PHHF(N ; k, v1u1 · · · vcuc , t) DHF(N ; k, w, t, v) DHHF(N ; k, v1u1 · · · vcuc , t, v) SHF(N ; k, w, {w1 , w2 , . . . , wt }) SHHF(N ; k, v1u1 · · · vcuc , {w1 , w2 , . . . , wt }) A basic construction follows: Lemma 4.7 If there exists a DHHF(N ; k, k1u1 · · · ukc c , t, v), then for every 1 ≤ i ≤ c, ui−1 ui+1 there exists a DHHF(N ; k − ⌊ kki ⌋, uu1 1 · · · ki−1 (ki − 1)1 kiui −1 ki+1 · · · kcuc , t, v) provided that ki ≥ v + 1. Proof. Consider a DHHF(N ; k, k1u1 · · · kcuc , t, v) and let r be a row that has ki symbols. The average number of times one of these ki symbols occurs in row r is kki , and hence some symbol σ occurs no more than ⌊ kki ⌋ times. In order to form a ui−1 ui+1 (ki − 1)ui ki+1 · · · kcuc , t, v), delete all columns that DHHF(N ; k − ⌊ kki ⌋, k1u1 · · · ki−1 contain σ in row r.  At first it appears that Lemma 4.7 is of little value, because a DHHF is needed to begin. However, a PHF(N ; k, w, t) is a DHF(N ; k, w, t, t), and a DHF(N ; k, w, t, v) is a DHHF(N ; k, wN , t, v); hence all constructions of perfect and distributing hash families provide input ingredients for Lemma 4.7. By eliminating one symbol from each of a number of rows, eliminating symbols from a single row, or a combination of the two, many DHHFs arise from a single DHF. The deletion of enough symbols in one row allows us to apply the following: Lemma 4.8 Whenever a DHHF(N ; k, k1u1 · · · kcuc , t, v) with ki < v exists, there is a ui−1 ui+1 ki+1 · · · kcuc , t, v). DHHF(N − ui ; k, k1u1 · · · ki−1 Proof. No row with fewer than v symbols can separate v classes, so we can remove all such rows without affecting the required separation.  Martirosyan and Tran Van Trung [98] essentially use a version of Lemma 4.8 in removing a row from a perfect hash family. They do not explore the extension to distributing hash families, and do not exploit the intermediate heterogeneous hash families that arise. Combining Lemma 4.8 with Lemma 4.7, we can manipulate both the number of rows and the number of symbols in each. Nevertheless we still require PHFs and DHFs to begin the process. Applying Lemma 4.7 to the DHF arising from an OA amounts to deleting points from the corresponding transversal design. Puncturing transversal designs has been extensively studied in another setting, the construction of mutually orthogonal latin squares via Wilson’s theorem; see [49,50] for a catalogue of structures in transversal designs that have been used in that context. Colbourn and Torres-Jiménez [59] consider only some of the more straightforward methods to puncture, as follows:

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Lemma 4.9 When q > s is a prime power, and the linear OA(q s ; s, q + 1, q) yields a DHF(M ; q s , q, t, v), there exist 1. a DHHF(M ; q s−1 α, q M −1 α1 , t, v) for v ≤ α ≤ q, and a DHF(M − 1; q s−1 (v − 1), q, s, v) (“one level”); 2. a DHHF(M ; q s−2 αβ, q M −2 α1 β 1 , t, v) for v ≤ α, β ≤ q (“two levels”); 3. a DHHF(M ; Sπ , q M −π (q −1)π , t, v) for 0 ≤ π ≤ M (“a spike”) – here S0 = q s and Sπ = Sπ−1 − ⌊ Sπ−1 q ⌋ for 1 ≤ π ≤ M ; M −1−π (q − 1)π α1 , t, v) for for v ≤ α ≤ q and 0 ≤ π ≤ 4. a DHHF(M ; Sπ,α , q S ⌋ M −1 (“a level and a spike”) – here S0,α = αq s and Sπ,α = Sπ−1,α −⌊ π−1,α q for 1 ≤ π ≤ M − 1. Proof. In each case we apply Lemma 4.7. For (1) delete q − α symbols in one row. For (2) further delete q − β symbols in another row. For (3) delete one symbol from each of π rows, and for (4) further delete q − α symbols in another row.  This is certainly not an exhaustive list, but it treats the majority of the applications in which we are interested. In determining Sπ in Lemma 4.9(3) and Sπ,α in Lemma 4.9(4), the structure of the OA(q s ; s, q + 1, q) is used in a naive manner. By explicitly constructing the OA and at each stage choosing a symbol to remove that minimizes the number of columns removed, we retain a number of columns Tπ or Tπ,α that is at least as large as Sπ or Sπ,α , respectively. Determining the largest number of columns that can be retained appears to be a challenging problem, but the greedy strategy employed here is an easy means to improve upon the simple argument of Lemma 4.9. See [59]. 5. Recursive Constructions of Hash Families Now we return to the construction of PHFs, and outline some recent results from [55,99]. A further definition is needed. A PHF(N m ; k, w, t) has matroshka type (N2 , N3 , . . . , Nt ) when, for each 2 ≤ m ≤ t, the first i=2 Ni rows form a PHF(N ; k, w, m). We require a variation on a DHF. Let (Γ, ⊙) be an abelian group of order n. Let A = (aiℓ ) be an N × k array with symbols from Γ. Let C be a set of m columns. Let (σ1 , . . . , σt ) be a t-tuple of elements of Γ, so that {σ1 , . . . , σt } contains at most t − 1 distinct values. Let C be a partition of {1, . . . , t} into m (possibly empty) classes {C1 , . . . , Cm } in such a way that (1) σi = σj only if i and j are in different classes; and (2) one class contains at least t+1−m elements. If there is a row ρ such that for every i, j with 1 ≤ i < j ≤ t, i ∈ Cx , and j ∈ Cy , we find that aρx ⊙ σi = aρy ⊙ σj , then A difference separates (C, C). When A difference separates all such choices for (C, C), it is a difference distributing hash family DDHF(N ; k, n, t, m). A DDHF(N ; k, n, t, m) has laminar type (M2 , . . . , Mm ) ℓ when, for 2 ≤ ℓ ≤ m, the first i=2 Mi rows form a DDHF(N ; k, n, t, ℓ). Colbourn and Ling [55] establish a generalization of the many constructions of [99, §4–6], so we consider the general result here. Theorem 5.1 Let w ≥ q. Suppose that there exist

1. a PHF(L1 ; k, w, t), a PHF(L2 ; κ, w, t), a PHF(N ; k, q, t − 1) with matroshka type (N2 , . . . , Nt−1 ), and

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2. a DDHF(M ; κ, q, t, t − 1) with laminar type (M2 , . . . , Mt−1 ).  Then there exists a PHF(L1 + L2 + Ng Mh ; kκ + w − q, w, t). g,h≥2 g+h≤t+1

Proof. Start with four arrays as follows.

1. A= (aij ) is an L1 × k array that is a PHF(L1 ; k, w, t). 2. B= (bij ) is an N × k array that is a PHF(N ; k, w, t − 1) with matroshka type (N2 , . . . , Nt−1 ). Let Bg be the submatrix containing the rows indexed from 1 + g g−1 i=2 Ni . i=2 Ni to 3. C= (cij ) is an M × κ array that is a DDHF(M ; κ, q, t, t − 1) with laminar type h (M2 , . . . , Mt−1 ). Let Ch be the submatrix containing the first j=2 Mj rows. 4. D= (dij ) is an L2 × k array that is a PHF(L2 ; κ, w, t).    t−1 t+1−g We form an L1 + L2 + g=2 Ng M × (kκ + w − q) array R. For conh h=2 venience we refer to columns of R using ordered pairs from ({1, . . . , k} × {1, . . . , κ}) ∪ {∞1 , . . . , ∞w−q }. To form R, we form certain arrays, each having kκ + w − q columns.

1. E= (eij ) is an array with L1 rows: Set ei,(j,ℓ) = aij for 1 ≤ i ≤ L1 , 1 ≤ j ≤ k, and 1 ≤ ℓ ≤ κ; set elements in columns indexed by {∞1 , . . . , ∞w−q } arbitrarily. 2. F= (fij ) is an array with L2 rows: Set fi,(j,ℓ) = di,ℓ for 1 ≤ i ≤ L2 , 1 ≤ j ≤ k, and 1 ≤ ℓ ≤ κ; set elements in columns indexed by {∞1 , . . . , ∞w−q } arbitrarily. t+1−g 3. For 2 ≤ g ≤ t − 1, Tg = (ti,(j,ℓ) ) is an array with h=1 Ng Mh rows: For every row ρ of Bg and every row r of Ct+1−g , form a row u of T with tu,(j,ℓ) = bρj ⊕ crℓ , where ⊕ is the addition defined for the DDHF. For each of the w − a unused symbols, adjoin a constant column consisting of one of that symbol.

Then vertically juxtapose E, F, and Tg for g = 2, . . . , t − 1 to form R. Choose any m ≤ t distinct columns {(j1 , ℓ1 ), . . . , (jt , ℓm )} from the kκ columns, and choose t − m from the columns indexed by {∞1 , . . . , ∞w−q }. If m = t and ℓ1 = · · · = ℓt , then {j1 , . . . , jt } are all distinct and the t columns are separated in E. If m = t and j1 = · · · = jt , then {ℓ1 , . . . , ℓt } are all distinct and the t columns are separated in F. Otherwise in each of the matrices Tg , each column among the last q − w is separated from every other column by every row. Let L be the set of distinct entries in {ℓ1 , . . . , ℓm } and J the set of distinct entries in {j1 , . . . , jm }. Suppose that J has g ′ distinct entries. Then 2 ≤ g ′ ≤ t−1 and L has at most t+1−g ′ distinct entries (when ja = jb , ℓa = ℓb ). Therefore the columns of J are separated by a row ρ of Bg for some g ≤ g ′ . We claim that a row of Tg separates the m columns. Set σi = bρ,ji for 1 ≤ i ≤ m and choose (σm+1 , . . . , σt ) arbitrarily. Because Ct+1−g is a DDHF(M ; κ, q, t, t + 1 − g), there is a row which difference separates the (at most t + 1 − g) columns of L for (σ1 , . . . , σt ).  Thus Tg separates {(j1 , ℓ1 ), . . . , (jm , ℓm )}. In order to apply Theorem 5.1, DDHFs are required. Basic constructions are given next.    Theorem 5.2 A DDHF(N ; q s−1 , q, t, t − 1) exists whenever q ≥ (s − 1) 2t − 1 . Indeed it has laminar type (M2 , . . . , Mt−1 ) for Mi = (s − 1)(t + 1 − i) for 2 ≤ i ≤ m.

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   ) elements of Fq , Proof. It suffices to show that for any set A of (s − 1)( 2t − t+1−m 2 the corresponding linear hash family A using only those polynomials of degree s with zero constant term forms a DDHF(N ; q s−1 , q, t, m). Two columns of A agree in at most s − 2 rows (in the orthogonal array they can agree in at most s − 1, but they surely agree in the row indexed by 0). Now choose any set C of m polynomials F1 , . . . , Fm , and choose elements {σ1 , . . . , σt } partitioned into m classes C = {C1 , . . . , Cv } of which one has at least t + 1 − m elements. Further, σi = σj only if i and j belong to different classes. Among the {σi } there are at most t − 1 distinct values; suppose without loss of generality that σi = σj , i ∈ C1 , and j ∈ C2 . We show that at least one of the rows indexed by A difference separates (C, C). The column indexed by F1 and that indexed by F2 agree in at most s − 2 rows; any element a row in which they   a ∈ A indexing  agree fails to separate. There remain at most 2t − t+1−m − 1 ways to select i and j in 2 different classes; for each, at most s − 1 elements fail to separate. Therefore at least one a ∈ A remains that difference separates.  Lemma 5.3 If there is a PHF(N1 ; k, v, t − 1) and a DDHF(N2 ; v, q, t, t − 1), then there exists a DDHF(N1 N2 ; k, q, t, t − 1). Proof. For 1 ≤ i ≤ v, replace each occurrence of symbol i in the PHF(N1 ; k, v, t − 1) by the ith column of the DDHF(N2 ; k, q, t, t − 1).  These lead to two corollaries of Theorem 5.1.  Corollary 5.4 [99, Theorem 5.1] Let w ≥ q ≥ 2t − 1 with q a prime power. Suppose that there  exist a PHF(L1 ; k, w, t) and a PHF(N ; k, q, t − 1). Then there exists a PHF(L1 + 2t − 1 N + 1; kq, w, t).  Proof. Set s = 2 and apply Theorem 5.2 to form a DDHF( 2t − 1; q, q, t). Take t its laminar type to be (M2 = 2 − 1, 0, . . . , 0). Take the matroshka type of the PHF(N ; k, q, t − 1) to be (N2 = N, 0, . . . , 0). Then form a PHF(L2 = 1; q, w, t). Apply Theorem 5.1 and omit the last w − q columns.   Corollary 5.5 [99, Theorem 4.1] Let q ≥ 2t − 1 be a prime power. Suppose that there   exist  a PHF(L1 ; k, q, t) and a PHF(N ; k, q, t − 1). Then there exists a PHF(L1 + t − 1 N + 1; kq, q, t). 2 Proof. Apply Corollary 5.4 with w = q.



Better ingredients for Theorem 5.1 can be found as well; we refer the interested reader to [55] for more details. With these definitions in hand, further DHHFs can be constructed as well [59]. Lemma 5.6 If a DHF(N ; k, w, t, v) with matroshka type (N2 , N3 , . . . , Nt ) exists, then there also exists a DHHF(N ; 2k, (2w)N −Nt wNt , t, v). Proof. Let A be a DHF(N ; k, w, t, v) with matroshka type (N2 , N3 , . . . , Nt ); partition j−1 its rows so that, for 2 ≤ j ≤ t, Aj consists of the Nj rows from row 1 + i=2 Ni to row j i=2 Ni . Form a matrix Bj from Aj on a disjoint set of symbols for 2 ≤ j < t. Form Fj by placing Aj and Bj side-by-side when 2 ≤ j < t, and placing At and At side-by-side

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when j = t. Then vertically juxtapose the arrays F2 , . . . , Ft to form an array E, which is an N × 2k array. Index columns of each array Aj and each array Bj by {1, . . . , k}, and index columns of E by {1, . . . , k} × {0, 1} in the natural way. Now choose t columns of E indexed by {(γ1 , i1 ), . . . , (γt , it )}, and a partition of these columns into v classes C1 , . . . , Cv . If |{γ1 , . . . , γt }| = t, there is a row of A that separates the classes C1 , . . . , Cv , and hence some row of E does as well. Otherwise |{γ1 , . . . , γt }| < t; form a new set of classes L1 , . . . , Lv by starting with C1 , . . . , Cv , and whenever γi = γj and i < j, remove (γj , ij ) from it class of the partition. Some row of A2 , A3 , or At−1 separates the classes L1 , . . . , Lv restricted to the first coordinates, because fewer than t distinct columns remain. Suppose that it is in Aj . Then in Fj there is a row that separates C1 , . . . , Cv because Aj and Bj share no symbols.  Our primary objective in treating these constructions here is to demonstrate that imposing structure on the ingredient arrays (for example, specifying matroshka and laminar types) enables us to avoid producing certain irrelevant rows. We observe the same phenomenon next, this time for covering arrays.

6. Constructing Covering Arrays using Hash Families Three constructions for covering arrays via column replacement were developed independently. One uses perfect hash families [15]. A second uses intersecting codes in the special case of binary covering arrays of strength three [118]. A third squares the number of factors using an array constructed from a Turán family [75,128]. The first has already been given in Theorem 3.1; here we explore how variants of hash families underlie the other two, and indeed many more, column replacement techniques. 6.1. Using Distributing Hash Families Colbourn [46] proves a general theorem: Theorem 6.1 Let k ≥ min(t, v). Suppose that there exist a DHF(M ; ℓ, k, t, min(t, v)) and a CA(N ; t, k, v) having ρ constant rows. Then a CA(ρ + (N − ρ)M ; t, ℓ, v) exists. Proof. Let A′ be a CA(N ; t, k, v) and let A be the (N −ρ)×k array obtained by removing the ρ constant rows. Let D be a DHF(M ; ℓ, k, t, min(t, v)) with symbols {1, . . . , k}. Let {aj : j = 1, . . . , k} be the columns of A. Form an (N − ρ)M × ℓ matrix by replacing the symbol j in D by the column aj . Then add ρ constant rows (one for each symbol in the constant rows deleted to form A from A′ ) to form the matrix E. It suffices to prove that E is a covering array of strength t. Fix a tuple C = (c1 , . . . , ct ) of t columns in E (equivalently, in D), and fix a t-way interaction T by selecting value ℓi for column ci for 1 ≤ i ≤ t. We must show that T is covered in E. If T is constant and contains a symbol of one of the ρ constant rows of A′ , it is covered in the ρ constant rows of E. Otherwise, the values (ℓ1 , . . . , ℓt ) partition C into nonempty classes C1 , . . . , Cw for w ≤ min(t, v), by placing ci and cj in the same class Cm if and only if ℓi = ℓj . There is a row (d1 , . . . , dℓ ) of D in which, for this partition, the entries ei = dci and ej = dcj are equal only if ci and cj belong to the same class Cm . Thus, on columns c1 , . . . , ct in E, there is an N × t subarray whose columns are ae1 , . . . , aet , in that order. The number z of distinct

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columns of A that are represented is at most min(t, v); because A is a covering array of strength t, it is also a covering array of strength z. Therefore the t-way interaction T is covered.  Applying Theorem 6.1 with ρ = 0 to a standardized CA, the constant row gives rise to M equal, constant rows. But applying the same theorem with ρ = 1, we obtain: Corollary 6.2 Let k ≥ min(t, v). Suppose that there exist a CA(N ; t, k, v) and a DHF(M ; ℓ, k, t, min(t, v)). Then a CA(1 + (N − 1)M ; t, ℓ, v) exists. Numerous constructions following the framework of Theorem 6.1 have been given. For binary arrays of strength three, combining Lemmas 4.4 and 3.4 (noting that T (3, 2) = 2) with Theorem 6.1 gives Theorem 7(ii) of Sloane [118]; this is a strengthening of earlier work in [28,112]. Implicitly, Sloane uses the fact that a certain binary intersecting code yields a DHF(N ; k, 2, 3, 2). Employing Corollary 6.2 in conjunction with Lemmas 4.4 and 3.4 establishes a somewhat improved result, reducing the number of rows in [118, Theorem 7(ii)] by 2x − 2: Theorem 6.3 Let q ≥ t ≥ 3 be a prime power and 1 ≤ x ≤ q. Suppose that a CA(N ; t, q, v) exists. If q + 1 > (x − 1)T (t, v) then a CA(1 + (N − 1)((x − 1)T (t, v) + 1); t, q x , v) also exists. Corollary 6.4 Let q ≥ 3 be a prime power and 1 ≤ x ≤ exists, a CA(1 + (N − 1)(2x − 1); 3, q x , 2) also exists.

q+2 2 .

When a CA(N ; 3, q, 2)

Proof. Take t = 3 and v = 2, so that T (t, v) = 2, to apply Theorem 6.3. We require that q + 1 > 2(x − 1), which is met when x ≤ q+2  2 . By Lemma 3.4, when q is a prime power there is a ΠA(q + 1; 2, q 2 , q). Applying Lemma 4.4 and Theorem 6.1, we recover a construction of Tang and Chen [128] and Boroday [22] for squaring the number of factors. Hartman [75] generalizes to non-primepower numbers of symbols by using Lemma 3.3 in place of Lemma 3.4 with t = 2; hence Hartman also squares the number of factors. In both cases, an improvement is obtained by applying Corollary 6.2 in place of Theorem 6.1, resulting in a savings of T (t, v) rows. We summarize this as follows. Theorem 6.5 If there are T (t, v) + 1 mutually orthogonal latin squares of order k, and a CA(N ; t, k, v) exists, a CA(1 + (N − 1)(T (t, v) + 1); t, k 2 , v) also exists. In [46], consequences for covering arrays are developed using the packing arrays described earlier. Further applications could employ packing arrays other than those produced by Lemmas 3.3 and 3.4. 6.2. Using Heterogeneous Hash Families Colbourn and Torres-Jiménez [59] improve upon Theorem 6.1 in two ways: judiciously choosing symbols on which to place the constant rows, and using heterogeneous hash families: Theorem 6.6 Suppose that there exist

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1. a CA(Ni ; t, ki , v) having ρi constant rows and ki ≥ t for 1 ≤ i ≤ c, and 2. a DHHF(M ; ℓ, k1u1 · · · kcuc , t, min(t, v)). c c Let χ = max(0, v − i=1 ui (v − ρi )). Then a CA(χ + i=1 ui (Ni − ρi ); t, ℓ, v) exists.

Proof. Let D be a DHHF(M ; ℓ, k1u1 · · · kcuc , t, min(t, v)). Partition the M rows of D into classes U1 , . . . , Uc so that, for 1 ≤ i ≤ c, class Ui contains exactly ui rows that each use (only) the symbols in {1, . . . , ki }. For 1 ≤ r ≤ M , choose Yr ⊆ {1, . . . , v} with |Yr | = +1 v − ρi when r ∈ Ui , and choose YM +1 with |YM +1 | = χ, so that ∪M r=1 Yr = {1, . . . , v}. For 1 ≤ r ≤ M , choose i so that r ∈ Ui , and let Br be a CA(Ni ; t, ki , v) whose ρi constant rows are on symbols {1, . . . , v} \ Yr . (Symbols can be renamed if necessary to place the constant rows on the desired symbols.) Then let Ar be the (Ni − ρi ) × ki array obtained by removing the ρi constant rows. Let {arj : j = 1, . . . , ki } be the columns of Ar for 1 ≤ r ≤ M . For each 1 ≤ r ≤ M , suppose that r ∈ Ui and form a (Ni − ρi ) × ℓ array Qr by replacing each occurrence of j in the rth row of D by the column arj . Form a χ × ℓ array S that contains a constant row for each symbol in cYM +1 . Then vertically juxtapose the arrays {Qr : 1 ≤ r ≤ M } and S to form a (χ + i=1 ui (Ni − ρi )) × ℓ matrix E. It suffices to prove that E is a covering array of strength t. Fix a tuple C = (c1 , . . . , ct ) of t columns in E (equivalently, in D), and fix a t-way interaction T by selecting value νj for column cj for 1 ≤ j ≤ t. We must show that T is covered in E. First consider the cases when T is constant, i.e. ν1 = · · · = νt = ψ. If ψ ∈ YM +1 , T is covered in S. Otherwise choose r so that ψ ∈ Yr , and consider the array Qr . Because Ar covers the constant s-tuple with all entries equal to ψ for every 1 ≤ s ≤ t, T is covered in Qr . Now consider cases when T is not a constant t-tuple. The values (ν1 , . . . , νt ) partition C into nonempty classes C1 , . . . , Cw for w ≤ min(t, v), by placing ca and cb in the same class if and only if νa = νb . Choose row r = (d1 , . . . , dℓ ) of D so that the entries ea = dca and eb = dcb are equal only if ca and cb belong to the same class; such a row exists because D is a DHHF. Choose i so that r ∈ Ui . On columns c1 , . . . , ct in Qr , there is an (Ni − ρi ) × t subarray whose columns are ar,e1 , . . . , ar,et , in that order. The number z of distinct columns of Ar that are represented is at most min(t, v); because Ar is a covering array of strength t, it is also a covering array of strength z. Therefore the t-way interaction T is covered.  Comparing with Theorem 6.1, even for DHFs an improvement is obtained when ρ < v: Corollary 6.7 Let k ≥ min(t, v). Suppose that a DHF(M ; ℓ, k, t, min(t, v)) and a CA(N ; t, k, v) having ρ constant rows both exist. Let χ = max(0, ρ − (M − 1)(v − ρ)). Then a CA(χ + (N − ρ)M ; t, ℓ, v) exists. Theorem 6.6 can exploit a library of small covering arrays of small covering arrays rather than a single one. Moreover, as in Theorem 2.2, Theorem 6.6 does not require that every ingredient array have many constant rows. In [59], most applications are given for DHHFs obtained from truncating the orthogonal array from the finite field. But they also give one example of the use of Lemma 5.6. There is a PHF(16; 172 , 17, 6) that has matroshka type (2, 2, 3, 4, 5) [55]. Lemma 5.6 gives a PHHF(16; 2 · 172 , 3411 175 , 6). For v ∈ {14, 15, 16} applying Theorem 6.6 with best

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values for CAN(6, 34, v) and a CAN(6, 17, v) yields improvements on the best known construction for CAN(6, 2 · 172 , v). 6.3. Using Partitioning Hash Families Consider a DHF(M ; 2d , 2, 3, 2) on symbols {0,1}. This cannot be used directly in Theorem 6.1, because the ‘covering   array’ required would have t > k. Nevertheless, replacing each occurrence of 0 by 01 and each occurrence of 1 by 10 , one obtains a 2M × 2d array on symbols {0, 1}. Adjoin the constant row with all elements 0, and the row with all elements 1, to form a (2M +2)×2d array. The result is a CA(2M +2; 3, 2d , 2). This is [118, Theorem 7(i)], which implicitly uses the same technique as Theorem 6.1 to remove constant rows prior to replication, and then replace them once at the conclusion.) Viewed as a column replacement construction, the hash family employed is partitioning and not just distributing. The use of partitioning hash families is appealing, because covering arrays of smaller strength can be used to make covering arrays of larger strength: Theorem 6.8 [46] Suppose that a PaHF(M ; ℓ, k, t, v) and a CA(N + ρ; v, k, v) with ρ constant rows both exist. Then a CA(ρ + N M ; t, ℓ, v) exists. Using a CA(2 + 2; 2, 2, 2) with 2 constant rows and a PaHF(M ; 2d , 2, 3, 2) from a binary intersecting code of length M and dimension d yields the theorem of Sloane. Other constructions of partitioning hash families appear to be needed to obtain further results. In [59], a variation on Theorem 6.6 is presented that requires additional properties of the DHHF but can save further rows. This can be seen as a generalization of the use of partitioning hash families. However, currently it does not appear to be fruitful in the construction of covering arrays. Theorem 6.9 Suppose that there exist 1. a CA(Ni ; t, ki , v) having ρi constant rows for 1 ≤ i ≤ c, and 2. a DHHF(M ; ℓ, k1u1 · · · kcuc , t, min(t, v)) for which, for every way to choose t columns, there is a row in which these t columns do not have all entries distinct. c Then a CA( i=1 ui (Ni − ρi ); t, ℓ, v) exists.

Proof. As in the proof of Theorem 6.6 form the arrays c Q1 , . . . , QM . Then vertically juxtapose the arrays {Qr : 1 ≤ r ≤ M } to form a ( i=1 ui (Ni − ρi )) × ℓ matrix E. It suffices to prove that E is a covering array of strength t. Fix a tuple C = (c1 , . . . , ct ) of t columns in E (equivalently, in D), and fix a t-way interaction T by selecting value νi for column ci for 1 ≤ i ≤ t. We must show that T is covered in E. When T is not constant, the argument is the same as in the proof of Theorem 6.6. Now consider the cases when T is constant, i.e. ν1 = · · · = νt = ψ. Choose any row r whose entries in columns c1 , . . . , ct are not all different and let kc1 , . . . , kct be the entries. Then z < t of them are distinct, so let {κ1 , . . . , κz } be the set of distinct entries. These index columns in Ar . Because Ar covers the constant z-tuple with all entries equal to ψ, T is covered in  Qr .

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7. Quilting Arrays Colbourn and Zhou [60] further improve on Theorem 6.6 using a variant of covering arrays, quilting arrays. The standard definition of covering array asks for all t-way interactions to be covered. The idea of covering only some of the t-way interactions has been examined in a number of contexts. Throughout this paper, for example, improvements often result by covering only the nonconstant t-way interactions. In [42] arrays of strength three are employed that cover all 3-way interactions except those in which all three symbols differ. Körner and Monti [88] relax the requirements in a different way; they study binary arrays of strength three in which at least 6 of the eight possible 3-way interactions arise in every set of three columns, but do not require that a specific set of 6 be covered. In a different direction, by restricting the sets of columns on which all interactions are to be covered, one encounters covering arrays on graphs [100,101] and variable strength covering arrays [40]. Here we consider restrictions in which all sets of t columns are treated similarly, but not all t-way interactions need to be covered. The species of a t-way interaction S = {(ci , νi ) : 1 ≤ i ≤ t} is the multiset {νi : 1 ≤ i ≤ t}; hence a species in general encompasses a number of specific t-way interactions. can be represented as a weak composition of t with v parts, and  (A species  there are t+v−1 species.) Often we are not concerned with the specific symbols used v−1 in defining the species. Then the family of a species is its orbit under the action of the symmetric group on v letters, and hence a family consists of a set of species, and by inheritance, a set of t-way interactions. (A family can be represented as a partition of t into at most v parts.) Let S be a set of species for t and v. An N × k array with v symbols is an Squilting array if every interaction whose species is in S is covered. The notation SQA(N ; t, k, v) is used for such an array when S contains interactions of strength at most t, and S-QAN(t, k, v) is the smallest N for which an S-QA(N ; t, k, v) exists. An SQA(N ; t, k, v) is equivalent to a CA(N ; t, k, v) when S contains all possible species of t-way interactions. We also employ a novel variant of hash families. Let S be the set of all multisets {ν1 , . . . , νt } with νi ∈ {1, . . . , v} for 1 ≤ i ≤ t. Let A be an M ×k array with v symbols. Define a function Φ with Φ : S → 2{1,...,M } . Then A is a Φ-separating hash family if for every S = {ν1 , . . . , νt } ∈ S and for every choice of t distinct columns (c1 , . . . , ct ), there is at least one row ρ ∈ Φ(S) in which, for 1 ≤ i < j ≤ t, row ρ has different symbols in columns ci and cj if νi = νj . Such an array is denoted by Φ-SHF(M ; k, v, t). Again we generalize to the heterogeneous case: A Φ–separating heterogeneous hash family ΦSHHF(M ; k, v1 · · · vM , t) contains at most vi symbols in the ith row for 1 ≤ i ≤ M , and satisfies the same separation condition. When Φ : S → 2{1,...,M } is specified, we define a vector (Ψ1 , . . . , ΨM ) so that Ψi = {S : i ∈ Φ(S)}. In words, Φ associates each t-way interaction with a set of rows of the array, while Ψi contains the t-way interactions thereby associated with the ith row. Theorem 7.1 Let t be a positive integer. Suppose that a Φ-SHHF(M ; k, k1 · · · kM , t) exM ists, and that a Ψi -QA(Ri ; t, ki , v) exists for each 1 ≤ i ≤ M . Then a CA( i=1 Ri ; t, k, v) exists. Proof. Let D be a Φ-SHHF(M ; k, k1 · · · kM , t). Form E by replacing each entry j in row i of D by the jth column of the Ψi -QA(Ri ; t, ki , v). It suffices to prove that E is a

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covering array of strength t. Fix a tuple C = (c1 , . . . , ct ) of t columns in E (equivalently, in D), and fix a t-way interaction T by selecting value νj for column cj for 1 ≤ j ≤ t. We must show that T is covered in E. Let W = Φ(T ), the set of rows that (together) separate T in D; then for some w ∈ W , T is separated in row w. Because T ∈ Ψw , it is covered in the Ψw -QA(Rw ; t, kw , v) and therefore also covered in E.  To recover Theorem 6.6, we equip the DHHF(M ; k, k1 · · · kM , t, min(t, v)) with a suitable function Φ. To do this, map every nonconstant multiset to the set of all rows; i constant multisets are treated differently. For 1 ≤ i ≤ M , let σi = min(v, j=1 ρj ) (using the notation of Theorem 6.6). Let σ0 = 0. We consider covering arrays on symbol set {0, . . . , v − 1}. For 1 ≤ i ≤ M , relabel symbols in a CA(Ni ; t, ki , v) so that it has ρi constant rows on symbols of {σi − ρi , . . . , σi − 1}, and delete these constant rows. M M However, when i=1 ρi > M (v − 1), retain χ = i=1 ρi − M (v − 1) constant rows in the CA(NM ; t, kM , v) on symbols v − χ, . . . , v − 1. Then for 1 ≤ i ≤ M − 1, Φ maps {ν1 , . . . , νt } with ν = ν1 = · · · = νt to row i if and only if ν < σi − ρi or ν ≥ σi . The result is a Ψi -QA(Ni − ρi ; t, ki , v). For the last row, Φ maps {ν1 , . . . , νt } with ν = ν1 = · · · = νt to row M if and only if ν < σM − ρM , ν ≥ σM , or v − χ ≤ ν < v. The result is a ΨM -QA(NM −ρM +χ; t, kM , v). Then Theorem 7.1 establishes the result of Theorem 6.6. Theorem 7.1 is more powerful, in that it employs much more than heterogeneity and constant rows. Nevertheless, its application is more technical as a result of the need for both Φ-separating hash families and suitable quilting arrays. We use DHHFs obtained by removing rows and/or columns from the transpose of an orthogonal array of strength s, but other constructions for DHHFs may prove useful here as well. For a species {ν1 , . . . , νt }, let τ ({ν1 , . . . , νt }) = |{(i, j) : νi = νj , 1 ≤ i < j ≤ t}|. Lemma 7.2 Suppose that a DHHF(M ; k, k1 · · · kM , t, v) exists for which every two different columns agree in at most s − 1 rows. Let S contain all multisets of size t from {1, . . . , v}. Suppose that Φ : S → 2{1,...,M } satisfies |{i : {ν1 , . . . , νt } ∈ Ψi , 1 ≤ i ≤ M }| > (s − 1) · τ ({ν1 , . . . , νt }). Then the DHHF is a Φ-SHF(M ; k, k1 · · · kM , t). Proof. Suppose to the contrary that some multiset S = {ν1 , . . . , νt } is not separated by the (at least) 1 + (s − 1) · τ ({ν1 , . . . , νt }) rows of Φ(S) in columns (c1 , . . . , ct ), where ci = cj whenever νi = νj . Two distinct columns agree in at most s − 1 rows, and hence each of the τ ({ν1 , . . . , νt }) pairs of unequal elements is separated by all but at most s − 1 rows. At most (s − 1) · τ ({ν1 , . . . , νt }) rows can fail to separate one or more of the pairs of columns (ci , cj ) with νi = νj . Hence at least one row of the DHHF separates all unequal pairs in {ν1 , . . . , νt }, which contradicts our hypothesis.  A number of DHHFs from Lemma 4.9 can be used in Lemma 7.2 [60]. We examine a few constructions for quilting arrays for use in Theorem 7.1. First we make an easy observation: Lemma 7.3 S-QAN(t, k, v) + S′ -QAN(t′ , k, v) ≥ (S ∪ S′ )-QAN(max(t, t′ ), k, v) ≥ SQAN(t, k, v). Proof. Vertically juxtaposing a S-QA(N1 ; t, k, v) and a S′ -QA(N2 ; t′ , k, v) yields a (S∪ S′ )-QA(N1 + N2 ; max(t, t′ ), k, v), and every (S ∪ S′ )-QA(N ; max(t, t′ ), k, v) is an SQA(N ; t, k, v). 

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Let S be a set of species on v symbols {0, . . . , v − 1}. For S ∈ S and 0 ≤ i < v, let ∂(S, i) be the empty set if S does not contain i, or the set of species obtained by removing exactly one i from S. Let ∂(S, i) = ∪S∈S ∂(S, i). Then we have an analogue of the usual derivation of covering arrays: v−1 Lemma 7.4 S-QAN(t, k, v) ≥ i=0 ∂(S, i)-QAN(t − 1, k − 1, v). For strength t there are t + 1 species when v = 2. For 1 ≤ i < t, let species Sit correspond to the composition (i, t − i); that is, it corresponds to the set of t-way interactions of weight i. (For convenience, define Sit = ∅ when i ∈ {0, . . . , t}.) Then define Sti = {Sit }, and when i < j define Sti,j = {Sit , . . . , Sjt }. One method of construction is essentially the construction of Tang and Chen [128]:   Lemma 7.5 Stℓ−(k−t),ℓ -QAN(t, k, 2) ≤ kℓ . Indeed Stℓ+1−α(k−t+1),ℓ -QAN(t, k, 2) ≤   α−1 k i=0 ℓ−i(k−t+1) when α is a positive integer.

Proof. For 0 ≤ i < α, form an array whose rows are all characteristic vectors of (ℓ − i(k − t + 1))-subsets of {1, . . . , k}. Then every t-way interaction of weight at least ℓ − i(k − t + 1) − (k − t) and at most ℓ − i(k − t + 1) specifies all but k − t of the column values; the remainder can be chosen to ensure that the row weight is ℓ − i(k − t + 1). Apply Lemma 7.3 to these α arrays to produce the desired array.  Another uses a Roux-type recursive construction in one specific case: Lemma 7.6 S42 -QAN(4, 2k, 2) ≤ S42 -QAN(4, k, 2) + CAN(3, k, 2). Proof. Let A be an N × k array that is a S42 -QAN(4, k, 2). Let B be an M × k array that is a CA(M ; 3, k, 2). Horizontally juxtapose two copies of A to form C, with columns indexed by {1, . . . , k} × {1, 2}. Horizontally juxtapose B and its complement to form D, with columns indexed similarly. Vertically juxtapose C and D to form a S42 -QA(N + M ; 4, 2k, 2). The verification is routine. Choose four columns {(ai , ci ) : 1 ≤ i ≤ 4}. If a1 , a2 , a3 , a4 are all distinct, they correspond in C to four distinct columns of A and hence the species in S42 are covered. If two of a1 , a2 , a3 , a4 are distinct, suppose without loss of generality that a1 = a3 , a2 = a4 , c1 = c2 = 1, and c3 = c4 = 2. All 4-way interactions of weight 2 are covered in C except for 0011, 0110, 1001, and 1100; these are all covered in D because every 2-way interaction is covered in B. If three of a1 , a2 , a3 , a4 are distinct, suppose without loss of generality that a1 = a4 , c1 = 1, and c4 = 2. In C, all 4-way interactions of weight 2 are covered except for 0011, 0101, 1010, and 1100. These are covered in D because in columns a1 , a2 , a3 of B are covered 001, 010, 101, and 110 to treat the case when c2 = c3 = 1; 000, 011, 100, and 111 to treat the case when c2 = 1 and c3 = 2; 011, 000, 111, and 100 to treat the case when c2 = 2 and  c3 = 1; and 010, 001, 110, and 101 to treat the case when c2 = c3 = 2. We expect that many other constructions for covering arrays can be extended and improved to produce bounds on binary quilting arrays. Colbourn and Zhou [60] use computational methods to make many examples, primarily using an heuristic postoptimization method [106].

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For quilting arrays in general, we aggregate species into families. As noted earlier, families for specific t and v correspond to partitions of t into at most v parts. Define v v-Ttx tocontain every family corresponding to a partition i=1 σi = t with σi ≥ 0 for 5 which 1≤i 0. (4) For all non-principal characters  e ,  Z(¦)^ > 0.

(5) For all , 0 < < "u − 1, N(8 ) − N(¨ ) + N(g ) > 0.

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Judicious choices of 8, ¨ and g as given in the next three theorems now yield new classes of "-ary perfect sequences. The first family works for " = 2, the second for any odd prime ", whereas the last one requires the said prime to be 3.

Theorem 6.15: Let " = 2 and  > 2 be an integer. Also let 2 be any integer with (2, ) = 1. Assume that  and 2 of opposite parity. Then ¦[1, −3, (2' + 1)] is a perfect sequence over (2u ) \{0}.

Remark 6.16: When " = 2 and (2, ) = 1, ¦[, V,— :] can be shown to be the binary Dillon and Dobbertin (2004) sequences. Kashyap (2005) has proved Theorem 6.15 independently using similar methods via Stickelberger combinatorics.

Theorem 6.17: Let p be an odd prime. Let d be an integer,  > 2. Let 2 be an Then integer with ("' + 1, "u − 1) = 2, or equivalently /(2, ) is odd. ¦[1, −2, "' + 1] is a perfect sequence over (p )\{0}.

Remark 6.18: It can be shown that odd prime "-ary perfect sequences of Dillon (2002), Helleseth and Gong (2002), Helleseth, Kumar and Martinsen (2001) arise as ¦[, ,!— :] of theorem 6.17 for each 2 with /(2, ) odd.

Theorem 6.19: Let " = 3 and  > 2 be an integer. Also let 2 be any integer with (2, ) = 1. Then ¦[1, −2, ½(3' + 1 )] is a perfect sequence over (3u )\{0}

(62)

If, furthermore,  is odd, ¦[1 + (3u − 1)/2, −2, ½(3' + 1) + (3u − 1)/2] is a perfect sequence over (3u )\{0} Remarks 6.20: (1) The two ternary perfect sequences ¦

[, ,

— ” ] a

Theorem 6.19 project to the same difference set.

and ¦

[:

—” 3 Д 3 Д ,  , : ] a a a

of

(2) Arasu, Dillon and Player (2010) have shown that these families prove the conjectures of Ludkovski and Gong (2001). (3) By focusing attention on the trace expansion of the perfect sequence in the last family with 2 = ( − 1)⁄2, Arasu, Dillon and Player (2010) show that it is equivalent to the Lin sequence (i.e. ¦ ~ R¥ R&'Zó:ó 3' + 1). Thus we obtain:

Corollary 6.21: The Lin Conjecture (1998) is true.

3 ^U

U , where  = 2 ∗

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Remarks 6.22: (1) Arasu, Dillon, Player (2010)) provide the first proof of this very important result. (2) Arasu, Dillon, Player (2010) do a lot more than what is stated in the above theorems. They also prove the inequivalence and compute the ranks in certain cases. We end this paper with the following: Question: Is there a more direct proof of the Lin Conjecture?

Acknowledgement The author wishes to thank John F. Dillon, Cunsheng Ding, Tor Helleseth, Dieter Jungnickel, Siu Lun Ma, Alexander Pott, and Bernhard Schmidt for their valuable suggestions.

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-172

Permutation decoding for codes from designs, finite geometries and graphs J.D. KEY Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, U.S.A. Abstract. Recent advances in technology have produced a requirement for new implementations of good error-correcting codes. Such applications of codes also require efficient encoding and decoding methods. The method of permutation decoding was first developed by MacWilliams in the early 60’s and can be used when a code has a sufficiently large automorphism group to ensure the existence of a set of automorphisms, called a PD-set, that has some specific properties. We describe here the method, and why it works, and give a short survey of permutation decoding using codes that arise from combinatorial structures such as graphs, designs and finite geometries, including some recent results in the search for PD-sets. Keywords. codes, designs, finite geometries, graphs

Introduction Permutation decoding was introduced by MacWilliams [42]. It involves finding a set of automorphisms of the code, called a PD-set, that acts in a certain way with respect to a known information set for the code. If such a set of automorphisms can be found, then a simple algorithm using this set can be followed to correct the maximum number of errors of which the code is capable. The method is described fully in MacWilliams and Sloane [43, Chapter 15] and also in Huffman [17, Section 8], where a survey of results up to the time of writing that chapter is given. We will describe the method and the algorithm in Section 2. We will give here a brief, but complete, description of permutation decoding, and discuss some recent results. In particular we will look at codes defined by classes of designs, graphs, or finite geometries where the automorphism group is known and large enough to allow permutation decoding or partial permutation decoding to be used. The implementation of this decoding method involves not only knowledge of the main parameters of the code, but also, of course, the automorphism group of the code, as well the ability to produce suitable information sets. This latter question leads to questions involving a set of basis vectors of the code, and in the case of codes from combinatorial structures that we concentrate on here, bases made up of incidence vectors of the blocks of the design, or a design associated

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with the structure. This particularly applies to codes from finite planes, for which there are only very partial answers. The question of the automorphism group of the code also arises, and the determination of when this is the same as the symmetry group of the combinatorial structure, or when it might be larger. The group of the code will be the same as the group of the design if the words in the code of weight the block size of the design are precisely the scalar multiples of the incidence vectors of the blocks of the design. This happens for a large class of graphs discussed here, and, of course, for finite projective planes. The results that are mentioned in the sections to follow do in many cases address these questions. In the sections to follow we first give some background material on designs, codes and graphs in Section 1. Section 2 contains a full description of permutation decoding and the notions of PD-sets and s-PD-sets. The remaining sections outline some of the known results for PD-sets from combinatorial structures. In many cases PD-sets for full error-correction are obtained, but in others it is not possible to use this method for full error-correction, due to a combinatorial lower bound on the size of a PD-set being larger than the size of the automorphism group. However in these case s-PD-sets can usually be found. In Section 6, as an illustration of the problems involved, we give sample proofs of three of the results quoted in the earlier sections, one for a class of graphs, one for desarguesian projective planes, and one for affine hyperplane designs.

1. Background and terminology The notation for designs and codes follows [1]. An incidence structure D = (P, B, J ), with point set P, block set B and incidence J is a t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and every t distinct points are together incident with precisely λ blocks. An incidence matrix M = [mi,j ] of D = (P, B, J ) with |B| = b is a b × v matrix with rows labelled by the blocks, columns by the points and mi,j = 1 if the ith block is incident with the j th point, and mi,j = 0 otherwise. A design is symmetric if v = b. The code CF (D) of the design D over the finite field F is the space spanned by the incidence vectors of the blocks over F . Equivalently, it is the row span of an incidence matrix for the design over F . If Q ⊆ P, then we denote the incidence vector 2 3 of Q by v Q , writing v P if Q = {P } where P ∈ P. Thus CF (D) = v B | B ∈ B , and is a subspace of F P . If F = Fp we write CF (D) = Cp (D). The p-rank of D, written rankp (D), is the dimension of Cp (D), i.e. the rank over Fp of an incidence matrix for D. The hull of a design with code C over Fp is C ∩ C ⊥ , written Hullp (D) or simply Hull(D). All the codes here will be linear codes, i.e. subspaces of the ambient vector space. If a code C over a field of order q is of length n, dimension k, and minimum weight d, we say that C is a [n, k, d]q code. A generator matrix for the code is a k × n matrix made up of a basis for C. The dual code C ⊥ is the orthogonal under the standard inner product, i.e. C ⊥ = {v ∈ F n |(v, c) = 0 for all c ∈ C}. A check matrix for C is a generator matrix for C ⊥ ; the syndrome of a vector y ∈ F n is Hy T . A code C is self-orthogonal if C ⊆ C ⊥ and is self-dual if C = C ⊥ . If c is a codeword then the support of c is the set of non-zero coordinate positions

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of c. A constant word in the code is a codeword, all of whose coordinate entries are either 0 or 1. The all-one vector will be denoted by j, and is the constant vector of weight the length of the code. Two linear codes of the same length and over the same field are equivalent if each can be obtained from the other by permuting the coordinate positions and multiplying each coordinate position by a non-zero field element. They are isomorphic if they can be obtained from one another by permuting the coordinate positions. Any code is isomorphic to a code with generator matrix in so-called standard form, i.e. the form [Ik | A]; a check matrix then is given by [−AT | In−k ]. The first k coordinates are the information symbols and the last n − k coordinates are the check symbols. An automorphism of a code C is any permutation of the coordinate positions that maps codewords to codewords. For any finite field Fq of order q, the set of points and r-dimensional subspaces of an m-dimensional projective geometry forms a 2-design which we will denote by P Gm,r (Fq ). Similarly, the set of points and r-dimensional flats of an m-dimensional affine geometry forms a 2-design, AGm,r (Fq ). The automorphism groups of these designs (and codes) are the full projective or affine semilinear groups, P Γ Lm+1 (Fq ) or AΓ Lm (Fq ), and are always 2-transitive on points. If q = pe where p is a prime, the codes of these designs are over Fp and are subfield subcodes of the generalized Reed-Muller codes: see [1, Chapter 5] for a full treatment. The dimension and minimum weight is known in each case: see [1, Theorem 5.7.9]. For a vector space V over a field F , if the dimension of V is n we will use ei , for i = 1, . . . , n to denote the standard basis for V . A translation by u ∈ V will be denoted by T (u), i.e. T (u) : x → x + u

(1)

for each x ∈ V . The group of translations of V will be denoted by T (V ) or simply T if the context is clear. The graphs, Γ = (V, E) with vertex set V and edge set E, discussed here are undirected with no loops, apart from the case where all loops are included, in which case the graph is called reflexive. If x, y ∈ V and x and y are adjacent, we write [x, y] for the edge in E that they define. If [xi , xi+1 ] for i = 1 to r − 1, and [xr , x1 ] are all edges of Γ, and the xi are all distinct, then the sequence written (x1 , . . . , xr ) will be called a closed path of length r for Γ. A graph is regular if all the vertices have the same valency. An adjacency matrix A of a graph with N vertices is an N × N matrix with entries aij such that aij = 1 if vertices vi and vj are adjacent, and aij = 0 otherwise. An incidence matrix of Γ is an N × |E| matrix B with bi,j = 1 if the vertex labelled by i is on the edge labelled by j, and bi,j = 0 otherwise. If Γ is regular with valency k, then the 1-(|E|, k, 2) design with incidence matrix B is called the incidence design of Γ. The neighbourhood design of a regular graph is the 1-design formed by taking the points to be the vertices of the graph and the blocks to be the sets of neighbours of a vertex, for each vertex, i.e. an adjacency matrix as an incidence matrix for the design. The line graph of a graph Γ = (V, E) is the graph L(Γ) with E as vertex set and where adjacency is defined so that e and f in E, as vertices, are adjacent in L(Γ) if e and f as edges of Γ share a vertex in Γ.

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The code of a graph Γ over a finite field F is the row span of an adjacency matrix A over the field F , denoted by CF (Γ) or CF (A). The dimension of the code is the rank of the matrix over F , also written rankp (A) if F = Fp , in which case we will speak of the p-rank of A or Γ, and write Cp (Γ) or Cp (A) for the code. It is also the code over Fp of the neighbourhood design. Similarly, if B is an incidence matrix for Γ, Cp (B) denotes the row span of B over Fp and is the code of the design with blocks the rows of B, in the case that Γ is regular. If M is an adjacency matrix for L(Γ) where Γ is regular of valency k, N vertices, e edges, then BB T = A + kIN and B T B = M + 2Ie ,

(2)

where A is an adjacency matrix, and B an incidence matrix, for Γ. When examining the codes from incidence matrices of graphs and adjacency matrices of their line graphs, the following result from [12] concerning weight-4 words in the dual code has been useful. We will refer to it in Section 6. Result 1 ([12]) Let Γ be a graph, G an incidence matrix for Γ, and [P, Q, R, S] a closed path in Γ. For any prime p, if C = Cp (G), then u = v [P,Q] + v [R,S] − v [P,S] − v [Q,R] ∈ C ⊥ .

(3)

For p odd, u ∈ Cp (L(Γ)). 2. Permutation decoding The decoding method termed permutation decoding involves finding a set of automorphisms of a code, called a PD-set. The method is described fully in MacWilliams and Sloane [43, Chapter 15] and Huffman [17, Section 8]. In [21] we extended the definition of PD-sets to s-PD-sets for s-error-correction, a term that is also used in [37,38]. Definition 1 If C is a t-error-correcting code with information set I and check set C, then a PD-set for C is a set S of automorphisms of C which is such that every t-set of coordinate positions is moved by at least one member of S into the check positions C. For s ≤ t an s-PD-set is a set S of automorphisms of C which is such that every s-set of coordinate positions is moved by at least one member of S into C. The algorithm for permutation decoding, once a PD-set has been found, is as follows: given a t-error-correcting [n, k, d]q code C with check matrix H in standard form. Thus the generator matrix G for C that is used for encoding has Ik as the first k columns, and hence as the information symbols. Any k-tuple v is encoded as vG. Suppose x is sent and y is received and at most t errors occur. Let S = {g1 , . . . , gm } be the PD-set.

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• Compute the syndromes H(ygi )T for i = 1, . . . , m until an i is found such that the weight of this vector is t or less. • Examine the information symbols in ygi , and obtain the codeword c that has these information symbols. • Decode y as cgi−1 . Note that this is valid since permutations of the coordinate positions correspond to linear transformations of F n , so that if y = x + e, where x ∈ C, then yg = xg + eg for any g ∈ Sn , and if g ∈ Aut(C), then xg ∈ C. That this method does correct t errors follows from the following result (proved in [17, Theorem 8.1]): Result 2 Let C be an [n, k, d]q t-error-correcting code. Suppose H is a check matrix for C in standard form, i.e. such that In−k is in the redundancy positions. Let y = c + e be a vector, where c ∈ C and e has weight ≤ t. Then the information symbols in y are correct if and only if the weight of the syndrome of y is ≤ t. Proof: Suppose C has generator matrix G in standard form, i.e. G = [Ik |A] and that the encoding is done using G, i.e. the data set x = (x1 , . . . , xk ) is encoded as xG. The information symbols are then the first k symbols, and the check matrix H is H = [−AT |In−k ]. Suppose the information symbols of y are correct. Then Hy T = HeT = eT , and thus wt(Hy T ) ≤ t. Conversely, suppose that not all the information symbols are correct. Then if e = e1 . . . en , and e′ = e1 . . . ek , e′′ = ek+1 . . . en , we assume that e′ is not the zero vector. Now use the fact that for any vectors wt(x + y) ≥ wt(x) − wt(y). Then T

T

wt(Hy T ) = wt(HeT ) = wt(−AT e′ + e′′ ) T

T

≥ wt(−AT e′ ) − wt(e′′ ) = wt(e′ A) − wt(e′′ )

= wt(e′ A) + wt(e′ ) − wt(e′ ) − wt(e′′ ) = wt(e′ G) − wt(e) ≥d−t≥t+1

which proves the result. ✷ There is a lower bound on the size of a PD-set (and one for an s-PD-set), due to Gordon [15] using a formula of Sch¨ onheim [46], and also proved in [17]: Result 3 ([15]) If S is a PD-set for a t-error-correcting [n, k, d]q code C, and r = n − k, then (

( ( ( ) ))) n−t+1 n n−1 |S| ≥ ... ... . r r−1 r−t+1

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This result can be adapted to s-PD-sets for s ≤ t by replacing t by s in the formula. In Gordon [15] and Wolfman [51] small PD-sets for the binary Golay codes were found. In Chabanne [6] abelian codes, i.e. ideals in the group algebra of an abelian group, are looked at using Gr¨obner bases, and the ideas of permutation decoding are generalized. In general it is rather hard to find these PD-sets, and obviously they need not even exist. Also the existence may depend on the chosen information set, and thus existence of a PD-set is not invariant under equivalence of codes. Note that PD-sets need not be sought, in general, for codes with minimum weight 3 or 4, since correcting a single error is, in fact, simply done by using syndrome decoding, because in that case multiples of the columns of the check matrix will give the possible syndromes. Thus the syndrome of the received vector need only be compared with the columns of the check matrix, by looking for a multiple. A simple argument yields that the worst-case time complexity for the decoding algorithm using an s-PD-set of size m on a code of length n and dimension k is O(nkm). Thus we want small PD-sets. Since the algorithm uses an ordering of the PD-set, good choices of the ordering of the elements can reduce the complexity. For example, we can find an s-PD-set Ss for each 0 ≤ s ≤ t such that S0 < S1 . . . < St and arrange the PD-set S in this order: S0 ∪ (S1 \ S0 ) ∪ (S2 \ S1 ) ∪ . . . ∪ (St \ St−1 ).

(4)

(Usually take S0 = {id}). A study of the complexity of the algorithm for some algebraic geometry codes is give in [19]. An interesting method of using anti-blocking sets, that is sometimes more efficient than that of PD-sets, is described in [38].

3. Cyclic codes and generalizations In her original paper, MacWilliams [42] developed a theory for finding PD-sets for cyclic codes. An [n, k, d]q code C is cyclic if whenever c = c1 c2 . . . cn ∈ C then every cyclic shift of c is in C. Thus the mapping τ ∈ Sn defined by τ : i → i + 1 for i ∈ {1, 2, . . . n}, is in the automorphism group of C, and τ n = 1. If a message c is sent and t errors occur, then if e is the error vector and if there is a sequence of k zeros between two of the error positions, then τ j for some j will move the sequence of zeros into the information positions, and thus all the errors will occur in the check positions. Thus < τ > will be a PD-set for C if k < nt .

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As shown in [42], if q is a number prime to the length n, then the map ρ : i → qi is also an automorphism of the cyclic code and in the normalizer N of < τ >. MacWilliams examines cases where N contains a PD-set. In [22, Lemma 7] the following, which generalizes this, was proved: Result 4 ([22]) Let C be a code with minimum distance d, I an information set, C the corresponding check set and P = I ∪ C. Let G be an automorphism group of C, and n the maximum of |O ∩ I|/|O|, where O is a G-orbit. If s = min(⌈ n1 ⌉ − 1, ⌊ d−1 2 ⌋), then G is an s-PD-set for C. It is important to note that this result is true for any information set. If the group G is transitive then |O| is the degree of the group and |O ∩ I| is the dimension of the code. This result is used to establish PD-sets in some of the classes of graphs and designs in the next sections, in particular when information sets are hard to find for general classes. Result 4 is applicable to codes from incidence matrices of connected regular graphs with automorphism groups transitive on edges: Result 5 ([8]) Let Γ = (V, E) be a regular graph of valency v with automorphism group A transitive on edges. Let M be an incidence matrix for Γ. If, for p a prime, C = Cp (M ) = [|E|, |V | − ǫ, v]p , where ǫ ∈ {0, 1, . . . , |V | − 1}, then any transitive subgroup K of A will serve as a PD-set for full error correction for C. This is used in the following section discussing PD-sets for some classes of graphs. 4. Codes from graphs In searching for PD-sets, suitable information sets need first to be found. Codes from some classes of graphs have large automorphism groups, so it was reasonable to consider some of these classes of graphs first. Notice that a code defined by a design or graph as outlined in Section 1 will have automorphism group at least that of the design or graph, and in some cases a larger automorphism group. We look here at codes from adjacency matrices and incidence matrices of classes of graphs, and find, where possible, information sets and PD-sets for these classes. In the following we frequently wish to include the identity element of the symmetric group Sn as a transposition in some set, and have used the notation (i, i), where 1 ≤ i ≤ n to denote it, where convenient. 4.1. Triangular graphs For any n, the triangular graph Tn is the line of the complete graph Kn ,  n graph  and is strongly regular with parameter set ( n−1 , 2(n − 2), n − 2, 4). Equivalently,   it is the uniform subset graph with vertex set the n2 2-subsets of a set of size n and adjacency defined by two 2-subsets being adjacent if the cardinality of their intersection in 1. The automorphism group of the graph is the symmetric group Sn , that being the automorphism group for Kn and hence, by [50], it is the group for the line graph Tn as well. Binary codes for Tn were examined in [16,49].

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In [26,45,13] the binary codes were examined for permutation decoding: Result 6 ([26] Theorem 1.1) For n ≥ 5, Tn the triangular graph, and C = C2 (Tn ) with the vertices I = {{1, n}, {2, n}, . . . , {n − 1, n}} in the first n − 1 positions:   1. C = [ n2 , n − 1, n − 1]2 code for n odd and, with I as the information set, S = {1G } ∪ {(i, n) | 1 ≤ i ≤ n − 1}

is a PD-set   for C of n elements in Sn ; 2. C = [ n2 , n − 2, 2(n − 2)]2 code for n even, and with I excluding {n − 1, n} as the information set, S = {1G } ∪ {(i, n) | 1 ≤ i ≤ n − 1} ∪ {[(i, n − 1)(j, n)]±1 | 1 ≤ i, j ≤ n − 2} is a PD-set for C of n2 − 2n + 2 elements in Sn . From [26], the automorphism group of the binary code of T (n) is also Sn for n ≥ 5, n = 6, since in the latter case the automorphism group of the code is larger. The computational complexity of the decoding by this method may be quite low, of the order n1.5 if the elements of the PD-set are appropriately ordered. The codes are low density parity check (LDPC) codes. Recall that, from Equation (2), if Mn is an incidence matrix for the complete graph Kn , and An is an adjacency matrix for Tn , then MnT Mn = An + 2I(

n n−1

).

Thus for binary codes, C2 (An ) ⊆ C2 (Mn ), and we are led to an examination of codes from Mn . (Note that in [30,12] it is shown that codes over other primes of the line graphs in this and similar cases will not yield interesting codes, since the minimum weight is at most 4. See Result 1) From Result 6 we see that for n odd C2 (Mn ) = C2 (Tn ) and for n even, C2 (Tn ) is the subcode spanned by the differences of two rows of Mn . In [30] the codes from the incidence matrix Mn of Kn over odd primes were examined: Result 7 ([30] Theorem 1.1) Let Cn be the p-ary code of an incidence matrix Mn for  the complete graph Kn where p is any odd prime and n ≥ 5. Then Cn is a [ n2 , n, n − 1]p code with information set I n = {[1, n], . . . , [n − 1, n], [1, 2]},

where [i, j] denotes the edge of Kn between the vertices i, j ∈ Ω.

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For n ≥ 6 the minimum words of Cn are the scalar multiples of the rows of Mn , and Aut(Cn ) = Sn . The set S = {(n, i)(1, j) | 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1} of elements of Sn , where (i, j) ∈ Sn is a transposition and (k, k) is the identity of Sn , is a PD-set of size n(n − 1) for Cn for the information set I n . For n ≥ 8, Cn has no words of weight d in the range n ≤ d ≤ 2n − 5.   Let En = ri − rj | ri , rj rows of Mn . Then for n ≥ 8, En is an [ n2 , n − 1, 2n−4]p code. For n ≥ 4, I ∗n = I n \{[n−1, n]} is an information set for En . For n ≥ 9, the minimum words of En are the scalar multiples of ri − rj , 1 ≤ i, j ≤ n, where ri , rj are rows of Mn . For n ≥ 7, S ∗ = {(n − 1, i)(n, j)(1, k) | 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n, 3 ≤ k ≤ n − 1}, is a PD-set of size n(n2 − 5n + 7) for En for the information set I ∗n . Note: This result holds for p = 2 as well, except that the dimension is n − 1 and one element must be removed from I n . 4.2. Lattice graphs The (square) lattice graph L2 (n) is the line graph of the complete bipartite graph Kn,n , and is strongly regular with parameters (n2 , 2(n − 1), n − 2, 2). The row span over F2 of an adjacency matrix (see also [49,16]) gives codes with parameters [n2 , 2(n−1), 2(n−1)]2 for n ≥ 5 with Sn ≀S2 as automorphism group. Information sets and PD-sets of size n2 in Sn × Sn were found in [35]. The vertex set of Kn,n is A ∪ B, where A = {a1 , . . . , an }, B = {b1 , . . . , bn } and the edges (points of the line graph) are the pairs [ai , bj ] where ai ∈ A, bj ∈ B. For σ, τ ∈ Sn , (σ, τ ) ∈ Sn × Sn acts on the points of L2 (n) by [ai , bj ](σ, τ ) = [aiσ , bjτ ]. Result 8 ([35] Theorem 1) For n ≥ 5, C = C2 (L2 (n)) = [n2 , 2(n − 1), 2(n − 1)]2 . The 2(n − 1) points {[ai , bn ] | 2 ≤ i ≤ n − 1} ∪ {[an , bi ] | 1 ≤ i ≤ n} are information symbols for which the set S = {((i, n), (j, n)) | 1 ≤ i ≤ n, 1 ≤ j ≤ n}

(5)

of permutations in Sn × Sn forms a PD-set of size n2 for C. Using the ideas of Equation (4), the time complexity can be reduced if the PD-set (sequence) is ordered as follows, from [48]:

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Result 9 ([48]) For n ≥ 5, C = C2 (L2 (n)) = [n2 , 2(n − 1), 2(n − 1)]2 , and information set {[ai , bn ]|2 ≤ i ≤ n − 1} ∪ {[an , bi ]|1 ≤ i ≤ n}, for 0 ≤ k ≤ t = n − 2, Sk = {((i, n), (j, n))|n − k ≤ i, j ≤ n} is a k-PD-set for C. The codes from incidence matrices for the lattice graph were also examined in [31]. Using the same notation for the graph as given above: Result 10 ([31] Theorem 1) Let Cn be the p-ary code of an incidence matrix Mn for the complete bipartite graph Kn,n where p is a prime and n ≥ 3. Then Cn is a [n2 , 2n − 1, n]p code with information set I n = {[ai , bn ] | 1 ≤ i ≤ n} ∪ {[an , bi ] | 1 ≤ i ≤ n − 1}. For n ≥ 3 the minimum words are the scalar multiples of the rows ri of Mn , and Aut(Cn ) = Sn ≀ S2 . The set S = {((n, i), (n, i)) | 1 ≤ i ≤ n}, of elements of Sn × Sn , where (i, j) ∈ Sn is a transposition if i = j, is a PD-set of size n for Cn using I n . Let En = ri − rj | ri , rj rows of Mn . Then for n ≥ 3 En is an [n2 , 2n − 2, 2n − 2]p code and the minimum words are the scalar multiples of the ri − rj . Further, I ∗n = I n \ {[a1 , bn ]} is an information set, and S ∗ = {((n, i), (n, j)) | 1 ≤ i, j ≤ n}, a PD-set of size n2 for En using I ∗n . Note that for p = 2, En is the binary code of the lattice graph L2 (n), and was covered originally in Result 8. 4.3. Line graphs of complete multi-partite graphs Similar results to those for the square lattice graph hold for the rectangular lattice graph L2 (m, n) [32], i.e. the line graph of the complete bipartite graph Km,n with n = m. The vertex set of Km,n is A∪B, where A = {a1 , . . . , am }, B = {b1 , . . . , bn } and the edges are the pairs [ai , bj ] where ai ∈ A, bj ∈ B. Result 11 ([32] Theorem 1) If C = C2 (L2 (m, n)) for 2 ≤ m < n, then C is • [mn, m + n − 2, 2m]2 for m + n even; • [mn, m + n − 1, m]2 for m + n odd

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with Sm × Sn as an automorphism group of C. The set I = {[ai , bn ] | 1 ≤ i ≤ m} ∪ {[am , bi ] | 1 ≤ i ≤ n − 1} is an information set for m + n odd, and I ∗ = I \ {[a1 , bn ]} is an information set for m + n even. Let S e = {((i, m), (j, n)) | 1 ≤ i ≤ m, 1 ≤ j ≤ m} ∪ {id}, S o = {((i, m), (i, n)) | 1 ≤ i ≤ m} ∪ {id}, be sets of permutations in Sm ×Sn . Then for 3 ≤ m < n, S e is a PD-set of m2 +1 elements for C for m + n even, and S o is a PD-set of m + 1 elements for C for m + n odd, using I as information symbols for m + n odd, and I ∗ for m + n even, and where id denotes the identity map. More generally, the binary codes of the line graphs L(Kn1 ,...,nm ) of the complete multi-partite graphs Kn1 ,...,nm , where ni = n for i = 1, . . . m, with automorphism group Sn ≀ Sm were considered in [33], and PD-sets were found for some classes, and s-PD-sets were found for all classes for some s. Writing the vertices of Kn,n,...,n as the ordered pairs (i, j) for 1 ≤ i ≤ m and 1 ≤ j ≤ n, the edges are [(i, j), (k, l)] where i = k, and these are the points of the line graph. The specific decoding sets can be found in [33]. Result 12 ([33] Theorem 1) If C = C2 (L(Kn,...,n )) is the binary code of the line graph of the complete multipartite graph Kn,...,n of nm vertices, where n ≥ 2, m ≥ 3, then • C is a [ 12 m(m − 1)n2 , mn − 2, 2n(m − 1) − 2 ]2 code for mn even; • C is a [ 12 m(m − 1)n2 , mn − 1, n(m − 1) ]2 code for mn odd.

Let I = {[(1, 1), (i, j)] | 2 ≤ i ≤ m, 1 ≤ j ≤ n}∪{[(1, i), (2, 1)] | 2 ≤ i ≤ n}\{[(1, 1), (m, n)]}, I ∗ = I ∪ {[(1, 2), (m, n)]}.

Then I is an information set for C if mn is even, and I ∗ is an information set for mn odd. Using these information sets 1. if n = 2 and m ≥ 3, C has a PD-set of size 16m2 − 8m; 2. if n = 3 and m ≥ 3 is odd, C has a PD-set of size 27m; 3. if m = 3 and n ≥ 3 is odd, C has a PD-set of size 2n3 . Furthermore, s-PD-sets of size N exist as follows: s < m/2, N = m; s < m, N = mn2 ; s < 3m/2, N = mn3 ; s < 2m, mn even, N = 4m2 n2 −2mn2 ; s < n/2, N = n for mn even, N = 2n for mn odd; s < n, N = n3 for mn even, N = 2n3 for mn odd.

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4.4. Uniform subset graphs If Ω is a set of size n, let P = Ω{3} , the set of subsets of Ω of size 3, be the vertex set of graphs Ai (n), for i = 0, 1, 2, with adjacency defined by two vertices (as 3-sets) being adjacent if the 3-sets have intersection of size i. The corresponding neighbourhood designs are denoted by Di (n), for i = 0, 1, 2. Properties of the binary codes of adjacency matrices of these graphs were established in [25]. Again Sn in its natural action acts as an automorphism group of the graphs and codes. The more interesting codes were examined for permutation decoding: Result 13([27] 1) Let C = C2 (D2 (n)) and n ≥ 7. Then for n odd,   Theorem C ⊥ = [ n3 , n−1 , n − 2] . 2 With 2

I = {{1, 2, n}, {1, 3, n}, . . . , {n−2, n−1, n}∪{{n−3, n−2, n−1}}\{{n−2, n−1, n}}

as information set, C ⊥ has a PD-set in Sn given by the following elements of Sn in their natural action on 3-subsets of Ω = {1, 2, . . . , n}: S = {(n, i)(n − 1, j)(n − 2, k) | 1 ≤ i ≤ n, 1 ≤ j ≤ n − 1, 1 ≤ k ≤ n − 2}. Note: The notation includes the convention (i, i) = 1, the identity element of Sn . Similar results hold for some of the other more interesting codes obtained in this way, but in some cases only partial decoding through s-PD-sets was possible: see [28]. Using the same notation: Result 14 ([28] Theorem 1) Let Ci (n) = C2 (Ai (n)) = C2 (Di (n)), for i = 0, 1 denote the code formed from the row span over F2 of an adjacency matrix for Ai (n).     For n = 4k, k ≥ 2, C0 (n)⊥ = [ n3 , n, n−1 2 ]2 with I = {{i, n − 1, n} | 1 ≤ i ≤ n − 2} ∪ {{n − 3, n − 2, n − 1}, {n − 3, n − 2, n}}

    as information set. For n ≡ 1 (mod 4), n ≥ 13, C1 (n)⊥ = [ n3 , n − 1, 2 n−2 2 ]2 and C1 (9)⊥ = [84, 8, 38]2 , with I \ {{n − 2, n − 1, n}} as information set. Taking the following elements of Sn , in their natural action on 3-subsets of Ω = {1, 2, . . . , n}: Σ1 = {(n, i) | 1 ≤ i ≤ n − 2} ∪ {ı}; Σ2 = {(n − 1, i) | 1 ≤ i ≤ n − 2} ∪ {ı}; Σ3 = {(n − 2, i) | 1 ≤ i ≤ n − 4} ∪ {ı}; Σ4 = {(n − 3, i) | 1 ≤ i ≤ n − 4} ∪ {ı}, where ı is the identity element of Sn , let Σ = Σ1 Σ2 Σ3 Σ4 . Then Σ is an s-PD-set for C0 (n)⊥ for s < ⌈n2 /6⌉ − 1, and for C1 (n)⊥ for s < ⌈n(n − 1)/6⌉ − 1.

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These graphs are particular cases of the class of uniform-subset graphs. A more general study of the binary codes of these graphs and the application of permutation decoding to the codes can be found in [13]. Ternary codes from the adjacency matrices of the graphs Ai (n) for i = 0, 1, 2 on 3-subsets were considered in [29] and permutation decoding can also be used for these, although the results are not published. 4.5. Hamming graphs Codes from adjacency and incidence matrices from the class of Hamming graphs H k (n, m) have been examined for permutation decoding in [34,10,9,11,7,12]. Here the Hamming graph H k (n, m), for n, k, m integers, 1 ≤ k < n, is the graph with vertices the mn n-tuples of Rn , where R is a set of size m, and adjacency defined by two n-tuples being adjacent if they differ in k coordinate positions. These are the graphs that occur in the Hamming association scheme: see [41, Chapter 30]. For example, the n-cube Qn is H 1 (n, 2) with R = F2 , and if k = 1 the standard notation H(n, m) is used. The automorphism group of H(n, m) is Sm ≀ Sn : see [4]. n From [34], using the notation for r ∈ Z and 0 ≤ r ≤ 2n −1, if r = i=1 ri 2i−1 is the binary representation of r, let r = (r1 , . . . , rn ) be the corresponding vector in Fn2 : Result 15 ([34] Theorem 1.1) For n even and n ≥ 8, let Tn = T {ti | 1 ≤ i ≤ n}, where T is the translation group of Fn2 , ti = (i, n) for i < n is a transposition in the symmetric group Sn , and tn is the identity map. Then Tn is a 3-PD-set of size n2n for the self-dual [2n , 2n−1 , n]2 code Cn from an adjacency matrix for H(n, 2), with the information set I = [0, 1, . . . , 2n−1 − 3, 2n − 2, 2n − 1]. In [10] it was shown that the same 2-PD-sets as found in [13] and 3-PDsets as found in [34] for C2 (H(n, 2)) for n even will work for C2 (H 2 (n, 2)) for n ≡ 0 (mod 4), n ≥ 8, although a different information set needs to be chosen. We do not have a formula for the minimum weight of C2 (H 2 (n, 2)), although we know it is 2 for n = 4, 8 for n = 8, and at least 12 for n = 12, by Magma. For n ≥ 4, with ti as in Result 15, let Pn = {ti | 1 ≤ i ≤ n − 1} ∪ {ι} and Tn = T Pn .

(6)

Since the translation group T is normalized by Sn , elements of the form T (w)ti T (u) are all in Tn , i.e. σ −1 T (u)σ = T (uσ −1 ), so that for transpositions t, tT (u) = T (ut)t. Let Pn∗ = {tn−1 , ι} and Tn∗ = T Pn∗ = T {tn−1 , ι}. Denote

(7)

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I 1 = {r | 0 ≤ r ≤ 2n−1 − 3} = {(r1 , . . . , rn−1 , 0) | ri ∈ F2 } \ {(0, 1, . . . , 1, 0), (1, . . . , 1, 0)}

C 1 = {r | 2n−1 ≤ r ≤ 2n − 1} \ {2n − 8, 2n − 7}

= {(r1 , . . . , rn−1 , 1) | ri ∈ F2 } \ {(0, 0, 0, 1, . . . , 1), (1, 0, 0, 1, . . . , 1)}

I 2 = {2n − 8, 2n − 7} = {(0, 0, 0, 1, . . . , 1), (1, 0, 0, 1, . . . , 1)} C 2 = {2n−1 − 2, 2n−1 − 1} = {(0, 1, . . . , 1, 0), (1, . . . , 1, 0)},

and I = I 1 ∪ I 2 and C = C 1 ∪ C 2 .

(8)

Result 16 ([10] Proposition 5) For n ≡ 0 (mod 4) and n ≥ 8, C = C2 (H 2 (n, 2)) = [2n , 2n−1 , d] where d ≥ 8. With I as in Equation (8) as information set, Tn∗ as in Equation (7) is a 2-PD-set of size 2n+1 for C, and Tn as in Equation (6) is a 3-PD-set of size n2n for C. The graph H(n, 3) and the related reflexive graph H ∗ (n, 3) (including all loops, i.e. by adding the identity matrix to an adjacency matrix for H(n, 3)) provide good binary codes from their adjacency matrices. Write D(n, 3) for the symmetric 1-(3n , 2n, 2n) design from an adjacency matrix for H(n, 3) and D(n, 3)∗ for the symmetric 1-(3n , 2n + 1, 2n + 1) design from an adjacency matrix for H ∗ (n, 3). Then, from [9], Result 17 ([9] Proposition 4) If n ≥ 4, then C = C2 (D(n, 3)) = [3n , 21 (3n − (−1)n ), 2n]2 and C ∗ = C ⊥ = C2 (D(n, 3)∗ ) = [3n , 21 (3n + (−1)n ), 2n + 1]2 . Further, C ∩ C ⊥ = {0} and the minimum words of C are the incidence vectors of the blocks of D(n, 3). For n ≥ 1, Aut(C2 (D(n, 3))) ∼ = S3 ≀ Sn ⊇ T ⋊ Sn , where T is the translation group on Vn = Fn3 . Using the natural ordering of the numbers 0 to 3n − 1 for n ≥ 3, if C = C2 (D(n, 3)) or C2 (D(n, 3)∗ ), and k = dim(C), then any consecutive set of k positions forms an information set for C. For n ≥ 3, if U = en−1 , en , then S = {T (u) | u ∈ U } is a 2-PD-set for C of size 9 for the information set from the natural ordering of the integers from 0 to 3n − 1. (As noted in Section 1, ei denotes a standard basis element for Vn = Fn3 ).) For the next result we need some notation: we write (i, n) for the transposition in Sn in its action on Vn , with (n, n) denoting the identity map. Further, di (a) will denote the diagonal matrix with all diagonal entries equal to 1, apart from the ith which will be a ∈ F× 3 . As before, T (u) will denote the translation by u ∈ Vn , T the translation group. Result 18 ([9] Proposition 5) For n ≥ 3, using the information set I for C = C2 (D(n, 3)) or C2 (D(n, 3)∗ ) obtained by rotating the naturally ordered numbers to the right by 3n−1 − 1 (thus starting with (1, 0, . . . , 0, 2) = 1 + 2(3n−1 )), the set

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S = T {(i, n)dn (a) | 1 ≤ i ≤ n, a ∈ F× 3} is a 3-PD-set for C of size 2n3n . Result 4 can be used for permutation decoding when a specific information set is not given. This is useful for codes from incidence matrices and has been applied to some cases, for example to the incidence designs and line graphs of H(n, 2): see [11]. Result 19 ([11] Proposition 19) For n ≥ 2 let C1 be the binary code obtained from the span over F2 of an adjacency matrix for the line graph L(H(n, 2)) of H(n, 2) and C2 the binary code spanned by an incidence matrix for H(n, 2). Then C1 ⊂ C2 , C1 is a [2n−1 n, 2n −2, 2(n−1)]2 code, and C2 is a [2n−1 n, 2n −1, n]2 code. For n ≥ 4, the minimum words of C1 and C2 are the rows of an adjacency, respectively incidence, matrix and the automorphism group of either code is T ⋊ Sn , where T is the translation group on Vn = Fn2 , and Sn the symmetric group of degree n acting on the n coordinate positions. Further, C1⊥ and C2⊥ have minimum weight 4, C1 ∩ C1⊥ ⊃ C2 ∩ C2⊥ , and C1 ∩ C1⊥ , respectively C2 ∩ C2⊥ , has dimension 2n−1 , respectively 2n−1 − 1, and minimum weight at most n2 for n even, or n(n − 1) for n odd. If E denotes the subgroup of T of translations by even-weight vectors, and g is an n-cycle in Sn , then Eg, regular of order 2n−1 n, is a ⌊ n2 ⌋-PD-set for C1 , a PD-set for C2 , and an (n − 1)-PD-set for Ci ∩ Ci⊥ , for i = 1, 2, for any information set. For the incidence matrices of the graphs H k (n, 2) for k ≥ 2, the following was proved in [8]. Result 20 ([8]Proposition 8) For   k ≥ 2, n ≥ 2k + 2, any transitive subgroup of Aut(H k (n, 2)) of degree 2n−1 nk acting on edges is a PD-set for any information set for the code Cp (Gkn (2)) from an incidence matrix Gkn (2) for H k (n, 2), where     1. for k odd, Cp (Gkn (2)) = [2n−1 nk , 2n − 1, nk ]p for all p, n k n−1 n n k 2. for k even, C2(G n (2)) = [2 k , 2 −2, k ]2 , and for p odd, Cp (Gn (2)) =    n n n−1 n [2 k , 2 , k ]p . Example 1 For k = 2, when n = q is a prime power, Sn will have sharply 2transitive subgroups. If H is any such then, then with T the translation group, T H has order 2n n(n − 1) and is easily seen to be transitive on the points of G 2n , and hence will be a PD-set. Similarly, if k = 3, n = q +1 where q is a prime-power, H a sharply 3-transitve group, we get PD-sets of size 2n n(n − 1)(n − 2).

4.6. Paley graphs If n is a prime power with n ≡ 1 (mod 4), the Paley graph,P (n), has Fn as vertex set and two vertices x and y are adjacent if and only if x − y is a non-zero square n−1 in Fn . It is a strongly regular graph with parameters (n, n(n−1) , n−1 2 4 − 1, 4 ). The row span over a field Fp of an adjacency matrix gives a good code (in fact, a quadratic residue code) if and only if p is a square in Fn .

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For any σ ∈ Aut(Fn ), if n = q e where q is prime, and a, b ∈ Fn with a a non-zero square, the set of mappings τa,b,σ : x → axσ + b form the automorphism group of the graph, of order 21 en(n − 1). It is not in general 2-transitive on vertices. Using Magma [5,3], it can be verified (see [20,40]) that for n ≥ 1697 and prime or n ≥ 1849 and a square, PD-sets cannot exist since the bound of Result 3 is bigger than the order of the group (using the square root bound for the minimum weight, and the actual minimum weight q + 1 when n = q 2 and q is a prime power). For the case where n is √ prime and n ≡ 1 (mod 8), the code of P (n) over Fp is C = [n, n−1 n, (the square-root bound) for p any prime dividing , d] where d ≥ p 2 n−1 . In [20] a 2-PD-set for C of size 6, and for the dual code, a 2-PD-set of size 4 10, was found for all n satisfying the stated conditions. Further results for this class of codes can be found in [40]. A general result was proved in [20,40], and used to find 2-PDsets for P (n) when n is a prime. Result 21 ([20] Theorem 1) Let C = [n, k, d]q be a cyclic code of prime length n over the field Fq of order q, where n ≡ 1 (mod 8), (n, q) = 1 and d ≥ 5. Label the coordinate positions 0, 1, . . . , n − 1 and suppose that 0, 1, . . . , k − 1 form the information symbols. Let τa,b : i → ai+b for a, b ∈ Fn where a is a nonzero-square and suppose that τa,b ∈ Aut(C) for all such a, b ∈ Fn . Then (1) if k =

n−1 2 ,

a 2-PD-set of size 6 for C is:

S = {τ1,b | b ∈ {0, k}} ∪ {τk,b | b ∈ {k, 2k, (2) if k =

n+1 2 ,

3k k , − 1}} 2 2

(9)

a 2-PD-set of size 10 for C is:

S = {τ1,b | b ∈ {0, 1, k, k−1, n−1}}∪{τk,b | b ∈ {0, k, k−1,

k − 1 3k − 1 , }}. 2 2 (10)

For the Paley graphs this gives: Result 22 ([20] Corollary 1) Let P (n) be the Paley graph of prime order n where n ≡ 1 (mod 8), and C = [n, n−1 2 ]p its code over Fp where p is a prime dividing n−1 . If the information set for C is I = {0, 1, . . . , k − 1}, where k = n−1 4 2 , then C has a 2-PD-set of size 6 as given by S in Equation (9). Result 23 ([20] Corollary 2) Let P (n) be the Paley graph of prime order n where n ≡ 1 (mod 8), and C ⊥ = [n, n+1 2 ]p the dual of its code C over Fp where p is a . If the information set for C ⊥ is I = {0, 1, . . . , k − 1}, where prime dividing n−1 4 n+1 ⊥ k = 2 , then C has a 2-PD-set of size 10 as given by S in Equation (10). Note: The lower bounds on the size of 2-PD-sets for the code and its dual of the Paley graph P (n) are 4 and 7, respectively, as follows immediately from Result 3. The sizes of 2-PD-sets in Result 22 and 23 are close to these bounds.

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5. Codes from finite geometries The finite geometries have, in general, far more structure than graphs, so their automorphism groups, AΓLn (Fq ) or P ΓLn (Fq ), are not as large, and in general will not accommodate PD-sets for full permutation decoding. However, in most cases s-PD-sets can be found for small s ≥ 2. The codes of the the designs from these affine and projective geometries are all from the family of generalized ReedMuller codes, including their subfield subcodes. This is all explained in detail in [1, Chapter 5], or [2]. We give a brief description of some members of the class below, but the main properties of these codes must be found elsewhere, and for example in [1, Chapters 5,6] 5.1. Generalized Reed-Muller codes Let q = pt , where p is a prime, and let V be the vector space Fm q of m-tuples, with standard basis. The codes will be q-ary codes with ambient space the function space FVq , with the usual basis of characteristic functions of the vectors of V . We can denote the elements f of FVq by functions of the m-variables denoting the coordinates of a variable vector in V , i.e. if x = (x1 , x2 , . . . , xm ) ∈ V, then f ∈ FVq is given by f = f (x1 , x2 , . . . , xm ) and the xi take values in Fq . Since aq = a for a ∈ Fq , the polynomial functions can be reduced modulo xqi − xi . Furthermore, every polynomial can be written uniquely as a linear combination of the q m monomial functions M = {xi11 xi22 . . . ximm | 0 ≤ ik ≤ q − 1, for 1 ≤ k ≤ m}. m For any such monomial the degree ρ is the total degree, i.e. ρ = k=1 ik and clearly 0 ≤ ρ ≤ m(q − 1). The generalized Reed-Muller codes are defined as follows (see [1, Definition 5.4.1]): Definition 2 Let V = Fm q be the vector space of m-tuples, for m ≥ 1, over Fq , where q = pt and p is a prime. For any ρ such that 0 ≤ ρ ≤ m(q − 1), the ρth -order generalized Reed-Muller code RFq (ρ, m) is the subspace of FVq (with basis the characteristic functions of vectors in V ) of all m-variable polynomial functions (reduced modulo xqi − xi ) of degree at most ρ. Thus RFq (ρ, m) = xi11 xi22 · · · ximm | 0 ≤ ik ≤ q − 1, for 1 ≤ k ≤ m,

m 

k=1

ik ≤ ρ.

These codes are thus codes of length q m and the codewords are obtained by evaluating the m-variable polynomials in the subspace at all the points of the ⊥ = vector space V = Fm q . From [1, Theorem 5.4.2] we know that RFq (ν, m) RFq (µ, m) for ν < m(q − 1) and where ν + µ + 1 = m(q − 1). For p prime, the code RFp ((m − r)(p − 1), m) is the p-ary code of the affine geometry design AGm,r (Fp ): see [1, Theorem 5.7.9]. For q = p = 2, the codes are the original Reed-Muller codes, written simply as R(m − r, m), this being the

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binary code of the the affine geometry design of points and r-dimensional flats of AGm (F2 ). The set of monomial functions of degree at most ν, B=

{xi11 xi22

. . . ximm

| 0 ≤ ik ≤ q − 1, for 1 ≤ k ≤ m,

m 

k=1

ik ≤ ν},

is an Fq -basis of RFq (ν, m). A subset S ⊆ V = Fq m will be an information set of the code if, and only if, the subspace of Fq S spanned by the restriction of B to S has dimension |B|. The following theorem from [22] holds for a wider class of codes spanned by monomials than the Generalized Reed-Muller codes: Result 24 ([22] Theorem 1) Let V = Fm q be the vector space of m-tuples, for m ≥ 1, over the finite field Fq of order q, where q = pt and p is a prime. Let α0 , . . . , αq−1 be the elements of Fq and let S = {[i1 , i2 , . . . , im ] | ik ∈ Z, 0 ≤ ik ≤ q − 1, 1 ≤ k ≤ m}. Let ≤ denote the partial order defined on S by [i1 , i2 , . . . , im ] ≤ [j1 , j2 , . . . , jm ] if and only if ik ≤ jk for all k such that 1 ≤ k ≤ m. Let X ⊆ S have the property that y ∈ X if y ∈ S and y ≤ x for some x ∈ X , and let C = xi11 xi22 · · · ximm | [i1 , i2 , . . . , im ] ∈ X . Then the set of vectors I = {(αi1 , . . . , αim ) | [i1 , i2 , . . . , im ] ∈ X } is an information set for C. m In particular, if X = {[i1 , i2 , . . . , im ] ∈ S | k=1 ik ≤ ν}, then I is an information set for RFq (ν, m), and if p is a prime, I = {(i1 , . . . , im ) | ik ∈ Fp , 1 ≤ k ≤ m,

m 

k=1

ik ≤ ν}

(11)

is an information set for RFp (ν, m) 5.2. Finite desarguesian planes If q = pe where p is prime, the code of the desarguesian projective plane of order )e + 1, q + 1]p . For the affine plane the code is q has parameters [q 2 + q + 1, ( p(p+1) 2 )e , q]p . The codes are subfield subcodes of the generalized Reed-Muller [q 2 , ( p(p+1) 2 codes (see [2]), and the automorphism groups are the semi-linear groups and doubly transitive on points. All these facts can be found in [1, Chapters 5,6]. Thus 2-PD-sets always exist. However, unlike the codes from graphs discussed in the preceding sections, it is not possible to obtain a general construction of PD-sets that will cover all members of this class of codes (i.e. for all q), since the bound of Result 3 for the size of a PD-set for error-correction using the full

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capability of the code is greater than the size of the group as q grows beyond a certain value: see Tables 1,2,3, [21]. For example, in the projective desarguesian case, P G2 (Fq ), when q is greater than the stated value, PD-sets for full errorcorrection cannot exist beyond the stated values of q (computations done using Magma [5] and GAP [14]): • • • • • • • •

q q q q q q q q

= p prime and p > 103; = 2e and e > 12; = 3e and e > 6; = 5e and e > 4; = 7e and e > 3; = 11e and e > 2; = 13e and e > 2; = pe for p > 13 and e > 1.

Similar results hold for the affine and for the dual codes. Thus it is not possible to give a general construction of PD-sets for this whole class of codes. However, s-PD-sets that apply to the whole class can be found for some small values of s ≥ 2. In [21] it was shown that both the code and its dual of any desarguesian projective plane will have 3-PD-sets no matter what information set I is chosen. To ensure that the code will correct three errors, we will take the order q ≥ 7; for the dual code, where the minimum weight in the case q = p prime is 2p, we need q ≥ 5. In general our bounds on the order relate to the error-correction capability of the code, which might not be the same as that of its dual. Result 25 ([21] Proposition 3.2) Let Π = P G2 (Fq ), where q = pe and p is a prime, C = Cp (Π) = [q 2 + q + 1, ( p(p+1) )e + 1, q + 1]p , and G = Aut(Π). Then 2 if q ≥ 7, a 3-PD-set can be found in G for C using any information set; similarly for q ≥ 5 for the dual code C ⊥ = [q 2 + q + 1, q 2 + q − ( p(p+1) )e , d⊥ ]p where 2 ⊥ q + p ≤ d ≤ 2q. If q ≥ 8, information sets exist for C such that 4-PD-sets can be found in G; similarly for C ⊥ for q ≥ 5. Similar results hold for the affine plane, but the information set is not arbitrary (see [21] for the properties required): Result 26 ([21] Proposition 3.3) Let π = AG2 (Fq ) where q = pe and p is a prime, )e , q]p , and G = Aut(π). Then if q ≥ 7, a 3-PD-set can C = Cp (π) = [q 2 , ( p(p+1) 2 be found in G for C. Similarly, for q ≥ 5, a 3-PD-set can be found in G for the )e , d⊥ ]p where q + p ≤ d⊥ ≤ 2q. dual code C ⊥ = [q 2 , q 2 − ( p(p+1) 2 For q = p a prime, using a Moorhouse [44] basis for the affine plane, 4-PD-sets were found for the affine case. Here the points of the plane AG2 (Fp ) are written as the ordered pairs, (a, b), for a, b ∈ Fp . Result 27 ([21] Proposition 3.4) Let π = AG2 (Fp ) where p is a prime and p ≥ 11,   , p] , and G = Aut(π). Then G contains a 4-PD-set for C = Cp (π) = [p2 , p+1 p 2 the code using information set

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I = {(i, j) | 0 ≤ i ≤ j ≤ p − 1},

(12)

C = {(i, j) | p − 1 ≥ i > j ≥ 0}.

(13)

and check set

The same result is true for p ≥ 5 for C ⊥ = [p2 , p2 − information set).

p+1 2

, 2p]p (using C as

5.3. Small 2-PD-sets in prime-order desarguesian planes It is clear that 2-PD-sets exist for any information set for the p-ary code of a desarguesian plane of order a power of p, since the group is 2-transitive. Since the smaller the size of an s-PD-set is, the more economical it will be for decoding purposes, it is desirable to find small 2-PD-sets inside the full group. In general this problem is not solved since information sets are not know in general. However, for prime order a Moorhouse [44] basis can be used to find an information set, and using this, in [21], the following sizes were obtained: • 2-PD-sets of 37 elements for desarguesian affine planes of any prime order p; • 2-PD-sets of 43 elements for desarguesian projective planes of any prime order p. Also 3-PD-sets for the code and the dual code in the affine prime case of sizes 2p2 (p − 1) and p2 , respectively, were found. In [21] the following general result was applied to planes of prime order: Result 28 ([21] Proposition 4.1) Let C = [n, k, d]q be a cyclic code of odd length n over the field Fq of order q, where k = n+1 2 , (n, q) = 1 and d ≥ 5. Label the coordinate positions 0, 1, . . . , n − 1 and take I = {0, 1, . . . , k − 1} for the information symbols. Let A = Aut(C) ≤ Sn , and let σ : i → i + 1 and µ : i → ≡ qi (mod n). If Z = σ and q ≡ ±1 (mod n), then S = Z ∪ µZ is a 2-PD-set of size 2n for C. Note: That µ ∈ Aut(C) is proved in MacWilliams [42]. Thus taking our informations positions to be consecutive positions defined by a cycle acting on the code, we will have the following: Result 29 ([21] Proposition 4.2) Let Π = P G2 (Fp ) where p ≥ 5 is a prime. Then C = Cp (Π) = [p2 + p + 1, n+1 2 , p + 1]p . Let Z be the cyclic group generated by a 2

Singer cycle and take I = {0, 1, . . . , p +p+2 − 1} for the information symbols, as 2 defined by S. Then, in the notation of Result 28 for σ and µ, where µ has order 3, Z ∪ µZ will form a 2-PD-set for C and Z will form a 2-PD-set for C ⊥ for p ≥ 3. We write here, for a translation in the affine group AGL2 (Fq ), τa,b : (x, y) → (x, y) + (a, b),

(14)

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for (x, y), (a, b) ∈ AG2 (Fq ). For a ∈ Fp and a = 0, define collineations of AG2 (Fp ): a ¯ : (x, y) → (ax, ay)

(15)

δ : (x, y) → (y, x)

(16)

for (x, y) ∈ AG2 (Fp ). Let Z = {¯ a | a ∈ Fp × } and T = {τa,b | 0 ≤ a, b ≤ p − 1}, the translation group of AG2 (Fp ). Using a Moorhouse basis for the code from the affine plane, the following was obtained in [21]: Result 30 ([21] Proposition 4.3) Let π = AG2 (Fp ) where p ≥ 5 is a prime, and C = Cp (π). Let n = ⌊(p + 1)/6⌋, and Y = {τun,−vn | 0 ≤ u, v ≤ 5}. Then, using I of Equation (12) as information set, Y is a 2-PD-set for C if p ≡ −1 (mod 6), and Y ∪ {τ1,1 } is a 2-PD-set for C if p ≡ 1 (mod 6), of size ≤ 37. Furthermore, (Y ∪ {τ1,1 })δ is a 2-PD-set of 37 elements for C ⊥ , using C of Equation (13) as information set, and where δ is defined in Equation (16). The analogue for the desarguesian projective planes of prime order was also obtained in [21]. First we define A = {(1, i, j) | 0 ≤ i, j ≤ p − 1}, A1 = {(1, i, j) | 0 ≤ i ≤ j ≤ p−1}, L = {(0, 1, i) | 0 ≤ i ≤ p−1} and P = (0, 0, 1) explicitly, and set A2 = A − A1 . Then we can take for an information set for Cp (P G2 (Fp )) the set I Π = {(1, i, j) | 0 ≤ i ≤ j ≤ p − 1} ∪ {(0, 0, 1)} = A1 ∪ {P },

(17)

and the corresponding check set will then be C Π = {(1, i, j) | p − 1 ≥ i > j ≥ 0} ∪ {(0, 1, i) | 0 ≤ i ≤ p − 1} = A2 ∪ L. (18) The element of P GL3 (Fq ) corresponding to the translation τa,b , we write τˆa,b

⎡

⎤ 1ab = ⎣0 1 0⎦ . 001

(19)

Result 31 ([21] Proposition 4.4) Let Π = P G2 (Fp ) where p ≥ 5 is a prime, and let C = Cp (Π). If n = ⌊(p + 1)/6⌋, let Yˆ = {ˆ τun,−vn | 0 ≤ u, v ≤ 5}, Yˆ0 = {ˆ τ0,0 , τˆ0,−(p−ε)/2 , τˆ−(p+ε)/2,−(p−ε)/2 , τˆ−(p−ε)/2,−p+ε }, where ε ∈ {−1, 1} and p ≡ ε (mod 6), and

⎤ ⎡ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎤ 1 0 0 010 010 100 100 σ0 = ⎣ 0 0 1 ⎦ , σ1 = ⎣ 0 1 1 ⎦ , σ2 = ⎣ 1 0 0 ⎦ , σ3 = ⎣ 0 −1 0 ⎦ , σ4 = ⎣ 0 0 1 ⎦ . 0 0 −1 100 001 010 010 ⎡

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Then, using the information set I Π of Equation (17), C has a 2-PD-set Yˆ ∪ Yˆ0 ∪ σ0 Yˆ0 ∪ {σ1 } in the case p ≡ −1 (mod 6) and Yˆ ∪ Yˆ0 ∪ σ0 Yˆ0 ∪ {σ1 , τˆ1,1 } in the case p ≡ 1 (mod 6), of size 42 and 43, respectively. Furthermore, using the information set C Π of Equation (18), the set (Yˆ ∪ {ˆ τ1,1 })σ0 ∪ {ι, σ2 , σ3 , τˆ1,1 σ3 , τˆ1,1 σ4 , τˆ−1,1 σ4 , σ4 , σ4 σ3 , τˆ1,0 σ4 } (where ι is the identity map) of size 46 is a 2-PD-set for C ⊥ . Also some specific 3-PD-sets in the affine case were found in [21]. Result 32 ([21] Proposition 4.5) Let π = AG2 (Fp ) where p is a prime, and let T be its translation group, Z and δ as defined above and in Equation (16). For p ≥ 7, T Z ∪ T Zδ is a 3-PD-set for the code C = Cp (π) using the information set of Equation (12), and for p ≥ 5, T is a minimal 3-PD-set for C ⊥ , using the information set of Equation (13). 5.4. Affine and projective geometry designs Information sets for the generalized Reed-Muller codes were found in [22] (see Result 24) and using these, 2-PD sets of size 2p3 for p ≥ 5 and 3-PD-sets of size p3 (p − 1)3 for p ≥ 7 were found in [23] for the p-ary codes from the 2-(p3 , p, 1) affine geometry designs of points and lines in 3-dimensional space over Fp , where p is a prime. Recall that it was mentioned in Section 5.1 that for p prime, the code RFp ((m−r)(p−1), m) is the p-ary code of the affine geometry design AGm,r (Fp ) of points and r-flats. So our code here has m = 3, r = 1, and is thus RFp (2(p−1), 3). Result 33 ([23] Theorem 1) Let D be the 2-(p3 , p, 1) design AG3,1 (Fp ) of points and lines in the affine space AG3 (Fp ), where p is a prime, and let C = Cp (D) = RFp (2(p − 1), 3). Then C is a [p3 , 61 p(5p2 + 1), p]p code with information set I = {(i1 , i2 , i3 ) | ik ∈ Fp , 1 ≤ k ≤ 3,

3 

k=1

ik ≤ 2(p − 1)}.

(20)

Let T be the translation group of AG3 (Fp ), let D be the group of invertible diagonal 3 × 3 matrices, and let Z be the group of scalar matrices. For each d ∈ Fp with d = 0, let µ(d) be the associated dilatation. Corresponding to the information set I, the code C has a 2-PD-set of the form T ∪ T µ(d) of size 2p3 for p ≥ 5 and for some d ∈ F∗p , and T D is a 3-PD-set for C of size p3 (p − 1)3 for p ≥ 7. For the 2-PD-set, we can choose d = (p − 1)/2. For q = p and r = m − 1, RFp ((m − r)(p − 1), m) = RFp (p − 1, m) = Cp (AGm,m−1 (Fp )), i.e. the code of the  affine  geometry design of points and m−1flats, or hyperplanes. Then |I| = m+p−1 . We have a general construction for m smaller 2-PD-sets for these designs for p ≥ 3 and m ≥ 3 (except for p = 3 when we will need m ≥ 4). This is from [22].

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Result 34 ([22] Proposition 2) Let C = Cp (AGm,m−1 (Fp )) = RFp (p−1, m) where p is a prime and p ≥ 3 and let Tm (Fp ) be the translation group. For the vector z = (1, 1, . . . , 1) ∈ Fm p let τ denote the translation by z and let Z = τ . Using the standard information set I = {(i1 , . . . , im ) | ik ∈ Fp , 1 ≤ k ≤ m,

m 

k=1

ik ≤ p − 1},

(21)

Z is a 2-PD-set of size p for C for m ≥ 3 and p ≥ 5, and for m ≥ 4 when p = 3. The general result concerning information sets (Result 24) for generalized Reed-Muller codes can be adapted to projective geometries over prime fields, and then partial PD-sets found as in the affine case. If I is an information set for Cp (AGm,m−1 (Fp )), then I ∗ ∪ {(0, . . . , 0, 1)} is an information set for Cp (P Gm,m−1 (Fp )), where I ∗ = {(1, x1 , . . . , xm ) | (x1 , . . . , xm ) ∈ I}. In Result 34, using the information set I of Equation (21), a 2-PD-set R = {τi | 0 ≤ i ≤ p−1} for Cp (AGm,m−1 (Fp )) was obtained, where τi is the translation τi | v → v + iz and z = (1, . . . , 1). Using the usual embedding of AGm (Fp ) into P Gm (Fp ), each τi corresponds to a collineation τˆi : (x0 , x1 , . . . , xm ) → (x0 , x1 + i, . . . , xm + i) of P Gm (Fp ). Let Z = {ˆ τi | 0 ≤ i ≤ p − 1}. We define two further collineations: µ : (x0 , . . . , xm−2 , xm−1 , xm ) → (x0 , . . . , xm−2 , xm , xm−1 ), ν : (x0 , x1 , . . . , xm−1 , xm ) → (x0 , x1 , . . . , xm−1 + xm , xm ), where the images are normalized further if necessary. Using these collineations we find a ‘small’ 2-PD-set for Cp (P Gm,m−1 (Fp )). Result 35 ([22] Proposition 6) For m ≥ 3, p ≥ 5, the set S = Z ∪ µZ ∪ {ν} of collineations of P Gm (Fp ) is a 2-PD-set of size 2p + 1 of the code Cp (P Gm,m−1 (Fp )) with respect to the information set I ∗ ∪ {(0, . . . , 0, 1)}. 5.4.1. Reed-Muller codes The first- and second-order Reed-Muller codes, R(1, m) and R(2, m), are binary codes with large minimum weight, being the codes of the affine geometry designs over F2 of points and (m − 1)-flats or (m − 2)-flats, respectively, and with the minimum words the incidence vectors of the blocks. In [24] the following was proved, extending results in [47]:

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Result 36 ([24] Theorem 1) Let V = Fm 2 and Ci = {v | v ∈ V, wt(v) = i} for 0 ≤ i ≤ m. Let T (u) denote the translation of V by u ∈ V , Am = {T (u) | u ∈ C0 ∪ C1 ∪ C2 ∪ Cm }, Bm = Am ∪ {T (u) | u ∈ C3 }, then 1. Am is an (m − 1)-PD-set of size 21 (m2 + m + 4) for R(1, m) for m ≥ 5 for the information set C0 ∪ C1 ; 2. Bm is an (m + 1)-PD-set of size 61 (m3 + 5m + 12) for R(1, m) for m ≥ 6 for the information set C0 ∪ C1 ; 3. Bm is an (m − 3)-PD-set of size 16 (m3 + 5m + 12) for R(2, m) for m ≥ 8 for the information set C0 ∪ C1 ∪ C2 . Some of these codes are also considered in [39]. 6. Examples In this section we illustrate three of the results described in the previous sections, showing how the process of examining the code of the graph or design for the existence of PD-sets or s-PD-sets was approached. Once the main parameters of the code have been established, a suitable information set needs to be found, and then the PD-set or s-PD-set itself found. For the latter process, Magma [5,3] was frequently used to help with the determination of such sets in the smaller cases (since all the examples are members of an infinite class), and then the general pattern established and verified theoretically. 6.1. Adjacency matrix of the Hamming graph Qn = H(n, 2) We refer here to the Result 15 from [34], and let An be an adjacency matrix for Qn = H(n, 2) so that with natural ordering of the vectors, for n ≥ 2 ! An−1 I An = . I An−1 We consider Cn = C2 (Qn ), and it follows that A2n = nI2n . Thus only the codes for n even will be of interest. It is shown in [34] that for n even, n ≥ 4, Cn is a [2n , 2n−1 , n]2 self-dual code with I = [0, 1, . . . , 2n−1 − 3, 2n − 2, 2n − 1] as an information set, using the notation as described prior to Result 15. We show that, for n even and n ≥ 8, Tn = {T (w)ti | w ∈ Fn2 , 1 ≤ i ≤ n}, where T (w) is the translation by w ∈ Fn2 , ti = (i, n) for i < n is a transposition in the symmetric group Sn , and tn is the identity map, is a 3-PD-set for Cn . If I is as above, the corresponding check set is C. We will write

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I 1 = [0, 1, . . . , 2n−1 − 3], C 1 = [2n−1 , 2n−1 + 1, . . . , 2n − 3] I 2 = [2n − 2, 2n − 1], C 2 = [2n−1 − 2, 2n−1 − 1] and a = 2n − 2 = (0, 1, . . . , 1, 1) , b = 2n − 1 = (1, 1, . . . , 1, 1)

A = 2n−1 − 2 = (0, 1, . . . , 1, 0) , B = 2n−1 − 1 = (1, 1, . . . , 1, 0) Notice that the points a and b are placed in I in order to have points and their complements in I since under any automorphism φ of the design, if vφ = w then vc φ = wc . Thus we have ac = 1 and bc = 0, Ac = 1 + 2n−1 , Bc = 2n−1 , and v + vc = b for any vector v ∈ P. Proof of Result 15: Let T = {x, y, z} be a set of three points in P. We need to show that there is an element in Tn that maps T into C. We consider the various possibilities for the points in T . If T ⊆ C then use ι. Thus suppose at least one of the points is in I and, by using a translation, suppose that one of the points, say z, is 0. If T ⊆ I, then T (2n−1 ) will work. Now we consider the other cases. 1. x ∈ I 1 , y ∈ C 1 Then there are ix , iy such that 2 ≤ ix , iy ≤ n − 1 such that x(ix ) = y(iy ) = 0. If ix = iy = i then T ti ⊆ I, unless yti ∈ {A, B}, so ti T (2n−1 ) will work unless yti ∈ {A, B}. If yti = A then y(1) = y(i) = 0, y(j) = 1 otherwise. If x(1) = 0 then t1 T (2n−1 ) will work. If x(i) = 1 then take any j = 1, i, n, and use T (2j−1 )ti T (2n−1 ). If yti = B, then y(i) = 0 and y(j) = 1 otherwise. Take any j = 1, i, n, and use T (2j−1 )ti T (2n−1 ). If x and y have no common zero, then if y = xc , so x + y = b, then use T (x)T (2n−1 ). If x(i) = y(i) = 1, where 1 ≤ i ≤ n − 1, then ti T (2n−1 − 1) can be used. 2. x ∈ I 1 , y ∈ C 2 (a) y = A: then x(i) = 0 for some 2 ≤ i ≤ n − 1, and ti T (2i−1 + 2n−1 ) will work. (b) y = B: then x(j) = 0 for some 1 ≤ j ≤ n − 1, and tj T (2j−1 + 2n−1 ) will work unless x(i) = 1 for all i = j, n. In the latter case, let 1 ≤ i ≤ n − 1, and i = j, then T (y)ti T (2i−1 + 2n−1 ) will work.

3. x ∈ I 2 , y ∈ C 1 (a) x = a: since y ∈ C 1 , there is a j such that 2 ≤ j ≤ n − 1 with y(j) = 0. If y(i) = 1 for i = j and 1 ≤ i ≤ n, or if y(1) = 0 and y(i) = 1 for i = j and 2 ≤ i ≤ n, then T (A) will work. If there is an i = j such that y(i) = y(j) = 0 where 2 ≤ i, j ≤ n − 1 then tj T (2n−1 ) can be used. (b) x = b: this follows exactly as in the x = a case except that in the first two cases for y use T (B) instead of T (A). 4. x ∈ I 2 , y ∈ C 2

(a) x = a, y = A: use T (a)t2 T (2n−1 ).

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(b) x = a, y = B: use tn−1 T (B). (c) x = b, y = A: use tn−1 T (B). (d) x = b, y = B: use t1 T (1 + 2n−1 ). 5. x, y ∈ C (a) x, y ∈ C 1 : if x + y = B then T (B) will work. Otherwise x(i) = y(i) for some i such that 1 ≤ i ≤ n − 1. Again T (B) will work unless x or y are (0, . . . , 0, 1) or (1, 0, . . . , 0, 1). If x = (0, . . . , 0, 1) then y(i) = 0 for some i such that 2 ≤ i ≤ n − 1. Then ti T (2n−1 ) can be used unless y(j) = 1 for all j = i, or y(1) = y(i) = 0 and y(j) = 1 for j = 1, i; in these cases ti T (2i−1 + 2n−1 ) can be used. The same arguments hold if x = (1, 0, . . . , 0, 1). (b) x ∈ C 1 , y ∈ C 2 i. y = A: since x ∈ C 1 , there is a j such that 2 ≤ j ≤ n − 1 with x(j) = 0. Then tj T (2j−1 + 2n−1 ) can be used unless y(i) = 1 for i = j and 1 ≤ i ≤ n, or if y(1) = 0 and y(i) = 1 for i = j and 2 ≤ i ≤ n. In these cases T (A)tk T (2k−1 + 2n−1 ), where k  1, j, 2 ≤ k ≤ n − 1, can be used. ii. y = B: exactly as in the case y = A, except that T (B) is used in the final cases. (c) x, y ∈ C 2 : x = A and y = B, and T (2n−2 + 2n−1 ) will work. This completes all the cases and proves the result. ✷ 6.2. Desarguesian projective planes We illustrate the proof of Result 25, and use the notation given there. For this we need a lemma, the proof of which is quite direct. Lemma 1 ([21] Lemma 3.1) If q = pe ≥ 5, where p is a prime, then  e e  p(p+1) e 2e > p + 2 and p − > pe + 2; 1. p(p+1) 2 e  2 e  + 1 > pe + 2 and p2e + pe + 1 − p(p+1) − 1 > pe + 2. 2. p(p+1) 2 2

Proof of Result 25: Note first that G is transitive on triangles and on collinear triples of points: see, for example, [18, Chapter 2]. For 3-PD-sets, let I denote an information set for C and C a check set, and let T = {P1 , P2 , P3 } be a set of three points. We first show that both I and C contain both triangles and sets of collinear triples. In fact, if a set of points in Π has no three points collinear, then it must be an arc in the plane and hence of size at most q + 2. Both I and C have size bigger than this by Lemma 1, so this is impossible. Also, neither I nor C can have all points collinear since this would restrict their size to q + 1. Thus both types of triples occur in both I and C. By transitivity, T can be mapped to the error positions by some member of G, in the case of C and in the case of C ⊥ .

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For 4-PD-sets, we need to consider sets of four points in Π. Such a set is either a quadrangle, or a point and three collinear points, or a set of four collinear points. Again taking I for the information set and C for the check set, using the lemma we see that both I and C contain 4-sets of the first two types. Since G is transitive on these types of 4-sets, we can always map such a 4-set to the check symbols. In the case of sets of four collinear points, we do not have transitivity. We have to ensure that C (for C) and I (for C ⊥ ) contains a representative of every orbit of G acting on collinear 4-sets. Since G is transitive on incident pointline pairs and since q ≥ 4, each line excluding an arbitrary point contains such representatives. We may choose I for C by starting with an information set for a corresponding affine plane and adding a point from the line at infinity. In this case C will contain a line excluding one point. Thus, C has a 4-PD-set in this case. Now let L be any line of P G2 (Fq ), let P1 ,. . . ,Pq+1 be the points of L, let P be a point off L and let Li be the line joining P to Pi , i = 1, . . . , q + 1. Then v L1 ,. . . ,v Lq+1 are independent and yield Iq+1 when restricted to the positions P1 ,. . . ,Pq+1 . Hence, we may choose I to contain P1 ,. . . ,Pq+1 . With the corresponding check set as the information set, C ⊥ has a 4-PD-set. ✷

6.3. Affine hyperplane designs We give the proof of Result 34 for Cp (AGm,m−1 (Fp )) = RFp (p − 1, m), the p-ary code of the design of points and hyperplanes of AGm (Fp ) where p is a prime. This uses the information set found in Result 24, Equation (11). Proof of Result 34: We need to show that any two vectors v and w can be movedby some multiple m of z into the check positions, C = {(i1 , i2 , . . . , im ) | ik ∈ Fp , k=1 ik > p − 1}. Notice that if, for a given prime p, we can prove this for m = t then it will follow for m ≥ t. To shorten the exposition, we will omit consideration of primes ≤ 11 and prove the result for p ≥ 13 and m = 3. This leaves m = 3 for the primes p = 5, 7 and 11 and m = 4 for p = 3. These involve a proliferation of subdivisions which need to be considered but no essential difficulty. We consider the various types of pairs of vectors (a, b, c) ∈ F3p and for each pair we write down an element k of Fp so that the corresponding element in Z that will move that pair into C. We can always translate such a pair of vectors into one of the form (a, b, c), (0, d, e). As membership of C depends only on the sum of the coordinates, we may assume that 0 ≤ a ≤ b ≤ c ≤ p−1 and 0 ≤ d ≤ e ≤ p−1. Let ℓ = ⌊p/3⌋ + 1. First, suppose d = e = 0. If p−1−a ≥ ℓ, let k = p−1−a unless b = c = a+1. In this case, if p − 2 − a ≥ ℓ let k = p − 2 − a, and if p − 1 − a = ℓ let k = 2ℓ + 1. If p − 1 − a < ℓ let k = p − 1. Next, suppose d = 0 and e = 0. If a + b + c > p + 2, let k = p − 1. If a + b + c ≤ p + 2 and p ≥ 11, let k = p − 1 − a unless b = c = a + 1. In this case, let k = p − 2 − a if p ≥ 13. Finally, suppose a, b and c are distinct and 0, d and e are distinct. If a+b+c > p + 2, let k = p − 1. Now suppose a + b + c ≤ p + 2. We may choose k = p − 1 − a

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if d ≤ a or if a < d and d + e ≥ 3a + 3. Now suppose additionally that a < d and d + e < 3a + 3. If e ≤ b let k = p − 1 − b and if b < e let k = p − e. This completes the proof for p ≥ 13 and m ≥ 3. ✷ 7. Conclusion We do not claim to give an exhaustive survey of all the known work to date on the discovery of PD-sets and s-PD-sets for codes associated with finite geometries or graphs, but we give a large sample of what has been achieved. Huffman [17] gives a survey up to the date of publication of his chapter in the Handbook of Coding Theory. Another survey of recent results is in [48]. A permutation decoding method linked to the method using PD-sets was established in [38,36]. It uses the idea of antiblocking sets, and works on the basis of finding a sufficient number of information sets to employ a decoding algorithm very similar to that used with PD-sets. This method can in fact be more efficient than that using PD-sets, as is pointed out in [36].

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E. F. Assmus, Jr and J. D. Key. Designs and their Codes. Cambridge: Cambridge University Press, 1992. Cambridge Tracts in Mathematics, Vol. 103 (Second printing with corrections, 1993). E. F. Assmus, Jr and J. D. Key. Polynomial codes and finite geometries. In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 1269–1343. Amsterdam: Elsevier, 1998. Volume 2, Part 2, Chapter 16. W. Bosma, J. Cannon, and C. Playoust. The Magma algebra system I: The user language. J. Symb. Comp., 24, 3/4:235–265, 1997. A. E. Brouwer, A. M. Cohen, and A. Neumaier. Distance-Regular Graphs. Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge 3, Band 18. Berlin, New York: Springer-Verlag, 1989. J. Cannon, A. Steel, and G. White. Linear codes over finite fields. In J. Cannon and W. Bosma, editors, Handbook of Magma Functions, pages 3951–4023. Computational Algebra Group, Department of Mathematics, University of Sydney, 2006. V2.13, http://magma.maths.usyd.edu.au/magma. Herv´ e Chabanne. Permutation decoding of abelian codes. IEEE Trans. Inform. Theory, 38:1826–1829, 1992. W. Fish, J. D. Key, and E. Mwambene. Binary codes from designs from the reflexive n-cube. Util. Math. (To appear 85 (2011)). W. Fish, J. D. Key, and E. Mwambene. Codes from the incidence matrices and line graphs of Hamming graphs H k (n, 2) for k ≥ 2. Adv. Math. Commun. (To appear). W. Fish, J. D. Key, and E. Mwambene. Codes, designs and groups from the Hamming graphs. J. Combin. Inform. System Sci., 34:169–182, 2009. No.1 – 4. W. Fish, J. D. Key, and E. Mwambene. Graphs, designs and codes related to the n-cube. Discrete Math., 309:3255–3269, 2009. W. Fish, J. D. Key, and E. Mwambene. Binary codes of line graphs from the n-cube. J. Symbolic Comput., 45:800–812, 2010. W. Fish, J. D. Key, and E. Mwambene. Codes from the incidence matrices and line graphs of Hamming graphs. Discrete Math., 310:1884–1897, 2010. Washiela Fish. Codes from uniform subset graphs and cyclic products. PhD thesis, University of the Western Cape, 2007.

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J.D. Key / Permutation Decoding for Codes from Designs, Finite Geometries and Graphs The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.4.12, 2008. http://www.gap-system.org. D. M. Gordon. Minimal permutation sets for decoding the binary Golay codes. IEEE Trans. Inform. Theory, 28:541–543, 1982. Willem H. Haemers, Ren´ e Peeters, and Jeroen M. van Rijckevorsel. Binary codes of strongly regular graphs. Des. Codes Cryptogr., 17:187–209, 1999. W. Cary Huffman. Codes and groups. In V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 1345–1440. Amsterdam: Elsevier, 1998. Volume 2, Part 2, Chapter 17. Daniel R. Hughes and Fred C. Piper. Projective Planes. Graduate Texts in Mathematics 6. New York: Springer-Verlag, 1973. David Joyner. Conjectural permutation decoding of some AG codes. ACM SIGSAM Bulletin, 39, 2005. No.1, March. J. D. Key and J. Limbupasiriporn. Permutation decoding of codes from Paley graphs. Congr. Numer., 170:143–155, 2004. J. D. Key, T. P. McDonough, and V. C. Mavron. Partial permutation decoding for codes from finite planes. European J. Combin., 26:665–682, 2005. J. D. Key, T. P. McDonough, and V. C. Mavron. Information sets and partial permutation decoding for codes from finite geometries. Finite Fields Appl., 12:232–247, 2006. J. D. Key, T. P. McDonough, and V. C. Mavron. Partial permutation decoding for codes from affine geometry designs. J. Geom., 88:101–109, 2008. J. D. Key, T. P. McDonough, and V. C. Mavron. Reed-Muller codes and permutation decoding. Discrete Math., 310:3114–3119, 2010. J. D. Key, J. Moori, and B. G. Rodrigues. Binary codes from graphs on triples. Discrete Math., 282/1-3:171–182, 2004. J. D. Key, J. Moori, and B. G. Rodrigues. Permutation decoding for binary codes from triangular graphs. European J. Combin., 25:113–123, 2004. J. D. Key, J. Moori, and B. G. Rodrigues. Binary codes from graphs on triples and permutation decoding. Ars Combin., 79:11–19, 2006. J. D. Key, J. Moori, and B. G. Rodrigues. Partial permutation decoding of some binary codes from graphs on triples. Ars Combin., 91:363–371, 2009. J. D. Key, J. Moori, and B. G. Rodrigues. Ternary codes from graphs on triples. Discrete Math., 309:4663–4681, 2009. J. D. Key, J. Moori, and B. G. Rodrigues. Codes associated with triangular graphs, and permutation decoding. Int. J. Information and Coding Theory, 1, No.3:334–349, 2010. J. D. Key and B. G. Rodrigues. Codes associated with lattice graphs, and permutation decoding. Discrete Appl. Math., 158:1807–1815, 2010. J. D. Key and P. Seneviratne. Binary codes from rectangular lattice graphs and permutation decoding. European J. Combin., 28:121–126, 2006. J. D. Key and P. Seneviratne. Codes from the line graphs of complete multipartite graphs and PD-sets. Discrete Math., 307:2217–2225, 2007. J. D. Key and P. Seneviratne. Permutation decoding for binary self-dual codes from the graph Qn where n is even. In T. Shaska, W. C. Huffman, D. Joyner, and V. Ustimenko, editors, Advances in Coding Theory and Cryptology, pages 152–159. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. Series on Coding Theory and Cryptology, 2. J. D. Key and P. Seneviratne. Permutation decoding of binary codes from lattice graphs. Discrete Math., 308:2862–2867, 2008. Hans-Joachim Kroll and Rita Vincenti. Antiblocking decoding. Discrete Appl. Math. (2010) To appear. Hans-Joachim Kroll and Rita Vincenti. PD-sets related to the codes of some classical varieties. Discrete Math., 301:89–105, 2005. Hans-Joachim Kroll and Rita Vincenti. Antiblocking systems and PD-sets. Discrete Math., 308:401–407, 2008. Hans-Joachim Kroll and Rita Vincenti. PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of PG(5,2). Discrete Math., 308:408–414,

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-202

Finite Groups, Designs and Codes

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J. MOORI 2 School of Mathematical Sciences, University of KwaZulu-Natal Pietermaritzburg 3209, South Africa Abstract. We will discuss two methods for constructing codes and designs from finite groups (mostly simple finite groups). This is a survey of the collaborative work by the author with J D Key and B Rorigues. Keywords. Designs, codes, simple groups, maximal subgroups, conjugacy classes

1. Introduction Error-correcting codes that have large automorphism groups whose properties are extensively studied can be useful in applications as the group can help in determining the code’s properties, and can be useful in decoding algorithms: see Huffman [14] for a discussion of possibilities, including the question of the use of permutation decoding by searching for PD-sets. We will discuss two methods for constructing codes and designs for finite groups (mostly simple finite groups). In the first method we discuss construction of symmetric 1-designs and binary codes obtained from the primitive permutation representations, that is from the action on the maximal subgroups, of a finite group G. This method has been applied to several sporadic simple groups, for example in [17], [21], [22], [26], [27], [28] and [29]. The second method introduces a technique from which a large number of non-symmetric 1-designs could be constructed. Let G be a finite group, M be a maximal subgroup of G and Cg = [g] = nX be the conjugacy class of G containing g. We construct 1 − (v, k, λ) designs D = (P, B), where P = nX and B = {(M ∩ nX)y |y ∈ G}. The parameters v, k, λ and further properties of D are determined. We also study codes associated with these designs. In Subsections 5.1, 5.2 and 5.3 we apply the second method to the groups A7 , P SL2 (q) and J1 respectively.

2. Terminology and notation Our notation will be standard, and it is as in [2] for designs and ATLAS [5] for groups. For the structure of finite simple groups and their maximal subgroups we follow the ATLAS notation. 1 AMS

Subject Classification (2000): 20D05, 05B05. from NATO, NRF and the University of KwaZulu-Natal are acknowledged.

2 Supports

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An incidence structure D = (P, B, I), with point set P, block set B and incidence I is a t-(v, k, λ) design, if |P| = v, every block B ∈ B is incident with precisely k points, and every t distinct points are together incident with ˜ = (P, B, I), ˜ where precisely λ blocks. The complement of D is the structure D t t ˜ I = P × B − I. The dual structure of D is D = (B, P, I ), where (B, P ) ∈ I t if and only if (P, B) ∈ I. Thus the transpose of an incidence matrix for D is an incidence matrix for Dt . We will say that the design is symmetric if it has the same number of points and blocks, and self dual if it is isomorphic to its dual. A t-(v, k, λ) design is called self-orthogonal if the block intersection numbers have the same parity as the block size. The code CF of the design D over the finite field F is the space spanned by the incidence vectors of the blocks over F . We take F to be a prime field Fp , in which case we write also Cp for CF , and refer to the dimension of Cp as the p-rank of D. If the point set of D is denoted by P and the block set by B, and if Q is any subset of P, then we will denote the incidence vector of Q by v Q . Thus   CF = v B | B ∈ B , and is a subspace of F P , the full vector space of functions from P to F . For any code C, the dual code C ⊥ is the orthogonal subspace under the standard inner product. The hull of a design’s code over some field is the intersection C ∩ C ⊥ . If a linear code over the finite field F of order q is of length n, dimension k, and minimum weight d, then we write [n, k, d]q to represent this information. If c is a codeword then the support of c, s(c), is the set of non-zero coordinate positions of c. A constant word in the code is a codeword all of whose coordinate entries are either 0 or 1. The all-one vector will be denoted by j, and is the constant vector of weight the length of the code. Two linear codes of the same length and over the same field are equivalent if each can be obtained from the other by permuting the coordinate positions and multiplying each coordinate position by a non-zero field element. They are isomorphic if they can be obtained from one another by permuting the coordinate positions. An automorphism of a code is any permutation of the coordinate positions that maps codewords to codewords. An automorphism thus preserves each weight class of C. A binary code with all weights divisible by 4 is said to be a doubly-even binary code. Terminology for graphs is standard: our graphs are undirected, the valency of a vertex is the number of edges containing the vertex. A graph is regular if all the vertices have the same valence, and a regular graph is strongly regular of type (n, k, λ, µ) if it has n vertices, valence k, and if any two adjacent vertices are together adjacent to λ vertices, while any two non-adjacent vertices are together adjacent to µ vertices. The groups G.H, G : H, and G· H denote a general extension, a split extension and a non-split extension respectively. For a prime p, pn denotes the elementary abelian group of order pn . If G is a group and M is a G-module, the socle of M , written Soc(M ), is the largest semi-simple G-submodule of M . It is the direct sum of all the irreducible G-submodules of M . Determination of Soc(V ) for each of the relevant full-space G-modules V = F n is highly desirable.

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3. Group Actions and Permutation Characters Suppose that G is a finite group acting on a finite set Ω. For α ∈ Ω, the stabilizer of α in G is given by Gα = {g ∈ G|αg = α}. Then Gα ≤ G and [G : Gα ] = |Δ|, where Δ is the orbit containing α. The action of G on Ω gives a permutation representation π with corresponding permutation character χπ denoted by χ(G|Ω). Then from elementary representation theory we deduce that Lemma 1 (i) The action of G on Ω is isomorphic to the action of G on the G/Gα , that is on the set of all left cosets of Gα in G. Hence χ(G|Ω) = χ(G|Gα ). (ii) χ(G|Ω) = (IGα )G , the trivial character of Gα induced to G. (iii) For all g ∈ G, we have χ(G|Ω)(g) = number of points in Ω fixed by g. Proof: For example see Isaacs [15] or Ali [1].  In fact for any subgroup H ≤ G we have χ(G|H)(g) =

k  |CG (g)| , |C H (hi )| i=1

where h1 , h2 , ..., hk are representatives of the conjugacy classes of H that fuse to [g] = Cg in G. Lemma 2 Let H be a subgroup of G and let Ω be the set of all conjugates of H in G. Then we have (i) GH = NG (H) and χ(G|Ω) = χ(G|NG (H). (ii) For any g in G, the number of conjugates of H in G containing g is given by χ(G|Ω)(g) =

m  i=1

k  |CG (g)| |CG (g)| = [NG (H) : H]−1 , |CNG (H) (xi )| |CH (hi )| i=1

where xi ’s and hi ’s are representatives of the conjugacy classes of NG (H) and H that fuse to [g] = Cg in G, respectively. Proof: (i) GH = {x ∈ G|H x = H} = {x ∈ G|x ∈ NG (H)} = NG (H). Now the results follows from Lemma 1 part (i).

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(ii) The proof follows from part (i) and Corollary 3.1.3 of Ganief [10] which uses a result of Finkelstien [7].  Remark 1 Note that χ(G|Ω)(g) = |{H x : (H x )g = H x }| = |{H x |H x

−1

gx

= H} =

|{H x |x−1 gx ∈ NG (H)}| = |{H x |g ∈ xNG (H)x−1 }| = |{H x |g ∈ (NG (H))x }|. Corollary 3 If G is a finite simple group and M is a maximal subgroup of G, then number λ of conjugates of M in G containing g is given by χ(G|M )(g) =

k  |CG (g)| , |C M (xi )| i=1

where x1 , x2 , ..., xk are representatives of the conjugacy classes of M that fuse to the class [g] = Cg in G. Proof: It follows from Lemma 2 and the fact that NG (M ) = M. It is also a direct application of Remark 1, since χ(G|Ω)(g) = |{M x |g ∈ (NG (M ))x }| = |{M x |g ∈ M x }|.  Let B be a subset of Ω. If B g = B or B g ∩ B = ∅ for all g ∈ G, we say B is a block for G. Clearly ∅, Ω and {α} for all α ∈ Ω are blocks, called trivial blocks. Any other block is called non-trivial. If G is transitive on Ω such that G has no non-trivial block on Ω, then we say G is primitive. Otherwise we say G is imprimitive. Remark 2 Classification of Finite Simple Groups (CFSG) implies that no 6transitive finite groups exist other than Sn (n ≥ 6) and An (n ≥ 8), and that the Mathieu groups are the only faithful permutation groups other than Sn and An providing examples for 4- and 5-transitive groups. Remark 3 It is well-known that every 2-transitive group is primitive. By using CFSG, all finite 2-transitive groups are known. The following is a well-known theorem that gives a characterisation of primitive permutation groups. Since by Lemma 1 the permutation action of a group G on a set Ω is equivalent to the action of G on the set of the left cosets G/Gα , determination of the primitive actions of G reduces to the classification of its maximal subgroups. Theorem 4 Let G be transitive permutation group on a set Ω. Then G is primitive if and only if Gα is a maximal subgroup of G for every α ∈ Ω. Proof: See Rotman [32]. 

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4. Method 1 Construction of 1-Designs and Codes from Maximal Subgroups: In this section we consider primitive representations of a finite group G. Let G be a finite primitive permutation group acting on the set Ω of size n. We can consider the action of G on Ω × Ω given by (α, β)g = (αg , β g ) for all α, β ∈ Ω and all g ∈ G. An orbit of G ¯ is an orbital, then Δ ¯ ∗ = {(α, β) : (β, α) ∈ Δ} ¯ on Ω × Ω is called an orbital. If Δ ¯ We say is also an orbital of G on Ω × Ω, which is called the paired orbital of Δ. ¯ is self-paired if Δ ¯ =Δ ¯ ∗. that Δ Now Let α ∈ Ω, and let Δ = {α} be an orbit of the stabilizer M = Gα of ¯ given by Δ ¯ = {(α, δ)g : δ ∈ Δ, g ∈ G} is an α. It is not difficult to see that Δ ¯ is a self paired orbital. Also orbital. We say that Δ is self-paired if and only if Δ note that the primitivity of G on Ω implies that M is a maximal subgroup of G. If M = Gα has only three orbits {α}, Δ and Δ′ on Ω, then we say that G is a rank-3 permutation group. Our construction for the symmetric 1-designs is based on the following results, mainly Theorem 5 below, which is the Proposition 1 of [17] with its corrected version in [18]: Theorem 5 Let G be a finite primitive permutation group acting on the set Ω of size n. Let α ∈ Ω, and let Δ = {α} be an orbit of the stabilizer Gα of α. If B = {Δg : g ∈ G} and, given δ ∈ Δ, E = {{α, δ}g : g ∈ G}, then D = (Ω, B) forms a 1-(n, |Δ|, |Δ|) design with n blocks. Further, if Δ is a self-paired orbit of Gα , then Γ = (Ω, E) is a regular connected graph of valency |Δ|, D is self-dual, and G acts as an automorphism group on each of these structures, primitive on vertices of the graph, and on points and blocks of the design. Proof: We have |G| = |ΔG ||G∆ |, and clearly G∆ ⊇ Gα . Since G is primitive on Ω, Gα is maximal in G, and thus G∆ = Gα , and |ΔG | = |B| = n. This proves that we have a 1-(n, |Δ|, |Δ|) design. Since Δ is self-paired, Γ is a graph rather than only a digraph. In Γ we notice that the vertices adjacent to α are the vertices in Δ. Now as we orbit these pairs under G, we get the nk ordered pairs, and thus nk/2 edges, where k = Δ. Since the graph has G acting, it is clearly regular, and thus the valency is k as required, i.e. the only vertices adjacent to α are those in the orbit Δ. The graph must be connected, as a maximal connected component will form a block of imprimitivity, contradicting the group’s primitive action. Now notice that an adjacency matrix for the graph is simply an incidence matrix for the 1-design, so that the 1-design is necessarily self-dual. This proves all our assertions.  Note that if we form any union of orbits of the stabilizer of a point, including the orbit consisting of the single point, and orbit this under the full group, we will

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still get a self-dual symmetric 1-design with the group operating. Thus the orbits of the stabilizer can be regarded as “building blocks”. Since the complementary design (i.e. taking the complements of the blocks to be the new blocks) will have exactly the same properties, we will assume that our block size is at most v/2. In fact this will give us all possible designs on which the group acts primitively on points and blocks: Lemma 6 If the group G acts primitively on the points and the blocks of a symmetric 1-design D, then the design can be obtained by orbiting a union of orbits of a point-stabilizer, as described in Theorem 5. Proof: Suppose that G acts primitively on points and blocks of the 1-(v, k, k) design D. Let B be the block set of D; then if B is any block of D, B = B G . Thus |G| = |B||GB |, and since G is primitive, GB is maximal and thus GB = Gα for some point. Thus Gα fixes B, so this must be a union of orbits of Gα .  Lemma 7 If G is a primitive simple group acting on Ω, then for any α ∈ Ω, the point stabilizer Gα has only one orbit of length 1. Proof: Suppose that Gα fixes also β. Then Gα = Gβ . Since G is transitive, there exists g ∈ G such that αg = β. Then (Gα )g = Gαg = Gβ = Gα , and thus g ∈ NG (Gα ) = N , the normalizer of Gα in G. Since Gα is maximal in G, we have N = G or N = Gα . But G is simple, so we must have N = Gα , so that g ∈ Gα and so β = α.  We have considered various finite simple groups, for example J1 ; J2 ; M c L; P Sp2m (q), where q is a power of an odd prime, and m ≥ 2; Co2 ; HS and Ru. For each group, using Magma [4], we construct designs and graphs that have the group acting primitively on points as automorphism group, and, for a selection of small primes, codes over that prime field derived from the designs or graphs that also have the group acting as automorphism group. For each code, the code automorphism group at least contains the associated group G. To aid in the classification, if possible, the dimension of the hull of the design for each of these primes were found. Then we took a closer look at some of the more interesting codes that arose, asking what the basic coding properties were, and if the full automorphism group could be established. It is well known, and easy to see, that if the group is rank-3, then the graph formed as described in Theorem 5 will be strongly regular. In case the group is not of rank 3, this might still happen, and we examined this question also for some of the groups we studied. A sample of our results for example for J1 and J2 is given below. Clearly the automorphism group of any of the codes will contain the automorphism group of the design from which it is formed. We looked at some of the codes that were computationally feasible to find out if the groups J1 and J¯2 formed the full automorphism group in any of the cases when the code was not the full vector space. We first mention the following lemma: Lemma 8 Let C be the linear code of length n of an incidence structure I over a field F. Then the automorphism group of C is the full symmetric group if and only if C = F n or C = F j⊥ .

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Proof: Suppose Aut(C) is Sn . C is spanned by the incidence vectors of the blocks of I; let B be such a block and suppose it has k points, and so it gives a vector of weight k in C. Clearly C contains the incidence vector of any set of k points, and thus, by taking the difference of two such vectors that differ in just two places, we see that C contains all the vectors of weight 2 having as non-zero entries 1 and −1. Thus C = F j⊥ or F n . The converse is clear.  Huffman [14] has more on codes and groups, and in particular, on the possibility of the use of permutation decoding for codes with large groups acting. See also Knapp and Schmid [25] for more on codes with prescribed groups acting. In [13] Haemers, Parker, Pless and Tonchev discuss a design and a code invariant under the simple group Co3 . We should also mention here that Tonchev [34] construct some binary linear codes using the adjacency matrices of the Hoffman-Singleton graph and the Higman-Sims graph. Most of the codes we looked at were too large to find the automorphism group, but we did find some of, through computation with Magma. Note that we could in some cases look for the full group of the hull, and from that deduce the group of the code, since Aut(C) = Aut(C ⊥ ) ⊆ Aut(C ∩ C ⊥ ). 4.1. J1 , J2 and Co2 In this subsection we give a brief discussion on the application of Method 1 to the sporadic simple groups J1 , J2 and Co2 . For full details the readers are referred to [17], [18], [19] and [27]. 4.1.1. Computations for J1 and J2 The first Janko sporadic simple group J1 has order 175560 = 23 ×3×5×7×11×19 and it has seven distinct primitive representations, of degree 266, 1045, 1463, 1540, 1596, 2926, and 4180, respectively (see Table 1 and [5,8]). For each of the seven primitive representations, using Magma, we constructed the permutation group and formed the orbits of the stabilizer of a point. For each of the non-trivial orbits, we formed the symmetric 1-design as described in Theorem 5. We took set of the {2, 3, 5, 7, 11} of primes and found the dimension of the code and its hull for each of these primes. Note also that since 19 is a divisor of the order of J1 , in some of the smaller cases it is worthwhile also to look at codes over the field of order 19. We also found the automorphism group of each design, which will be the same as the automorphism group of the regular graph. Where computationally possible we also found the automorphism group of the code. Conclusions from our results are summarized below. In brief, we found that there are 245 designs formed in this manner from single orbits and that none of them is isomorphic to any other of the designs in this set. In every case the full automorphism group of the design or graph is J1 . In Table 2, the first column gives the degree, the second the number of orbits, and the remaining columns give the length of the orbits of length greater than 1, with the number of that length in parenthesis behind the length in case there is more than one of that length. The pairs that had the same code dimensions occurred as follows: for degrees 266, 1045 and 1596, there were no such pairs; for

J. Moori / Finite Groups, Designs and Codes No.

Order

Index

Structure

Max[1] Max[2]

660

266

P SL(2, 11)

168

1045

23 :7:3

Max[3]

120

1463

Max[4]

114

1540

2 × A5

Max[5]

110

1596

11:10

Max[6]

60

2926

Max[7]

42

4180

D6 × D10

209

19:6

7:6

Table 1. Maximal subgroups of J1

Degree

#

length

266

5

132

110

1045

11

168(5)

56(3)

28

8

1463

22

120(7)

60(9)

20(2)

15(2)

1540

21

114(9)

57(6)

38(4)

19

1596

19

110(13)

55(2)

22(2)

11

2926

67

60(34)

30(27)

15(5)

4180

107

42(95)

21(6)

14(4)

12

11 12

7

Table 2. Orbits of a point-stabilizer of J1

degree 1463 there were two pairs, both for orbit size 60; for degree 1540, there were two pairs, for orbit size 57 and 114 respectively; for degree 2926 there was one pair for orbit size 60; for degree 4180 there were 12 pairs, for orbit size 42. In summary then, we have the following: Proposition 9 If G is the first Janko group J1 , there are precisely 245 nonisomorphic self-dual 1-designs obtained by taking all the images under G of the non-trivial orbits of the point stabilizer in any of G’s primitive representations, and on which G acts primitively on points and blocks. In each case the full automorphism group is J1 . Every primitive action on symmetric 1-designs can be obtained by taking the union of such orbits and orbiting under G. We tested the graphs for strong regularity in the cases of the smaller degree, and did not find any that were strongly regular. We also found the designs and their codes for some of the unions of orbits in some cases. We found that some of the codes were the same for some primes, but not for all. The second Janko sporadic simple group J2 has order has order 604800 = 27 × 33 × 52 × 7, and it has nine primitive permutation representations (see Table 3), but we did not compute with the largest degree. Thus our results cover only the first eight. Our results for J2 are different from those for J1 , due to the existence of an outer automorphism. The main difference is that usually the full automorphism group is J¯2 , and that in the cases where it was only J2 , there would be another orbit of that length that would give an isomorphic design, and which, if the two orbits were joined, would give a design of double the block size and

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automorphism group J¯2 . A similar conclusion held if some union of orbits was taken as a base block. No.

Order

Index

Max[1] Max[2] Max[3] Max[4] Max[5] Max[6]

6048 2160 1920 1152 720 600

100 280 315 525 840 1008

Max[7] Max[8] Max[9]

336 300 60

1800 2016 10080

Structure P SU (3, 3) 3. P GL(2, 9) 21+4 :A5 22+4 :(3 × S3 ) A4 × A5 A5 × D10 P SL(2, 7):2 52 :D12 A5

Table 3. Maximal subgroups of J2

From these eight primitive representations, we obtained in all 51 nonisomorphic symmetric designs on which J2 acts primitively. Table 4 gives the same information for J2 that Table 2 gives for J1 . The automorphism group of the design in each case was J2 or J¯2 . Where J2 was the full group, there is another copy of the design for another orbit of the same length. This occurred in the following cases: degree 315, orbit length 32; degree 1008, orbit lengths 60, 100 and 150; degree 1800, orbit lengths 42, 42, 84 and 168; degree 2016, orbit lengths 50, 75, 75, 150, 150, and 300. We note again that the p-ranks of the design and their hulls gave an initial indication of possible isomorphisms and clear non-isomorphisms, so that only the few mentioned needed be tested. This reduced the computations tremendously. We also found three strongly regular graphs (all of which are known: see Brouwer [6]): that of degree 100 from the rank-3 action, of course, and two more of degree 280 from the orbits of length 135 and 36, giving strongly regular graphs with parameters (280,135,70,60) and (280,36,8,4) respectively. The full automorphism group is J¯2 in each case. We have not checked all the other representations but note that this is the only one with point stabilizer having exactly four orbits. Note that Bagchi [3] found a strongly regular graph with J2 acting. Degree

#

100

3

length 63

36

280

4

135

108

36

315

6

160

80

32(2)

10

525

6

192(2)

96

32

12

840

7

360

240

180

24

20

15

1008

11

300

150(2)

100(2)

60(2)

50

25

12

1800

18

336

168(6)

84(3)

42(3)

28

21

14(2)

2016

18

300(2)

150(6)

75(5)

50(2)

25

15

Table 4. Orbits of a point-stabilizer of J2 (of degree ≤ 2016)

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In each of the following we consider the primitive action of J2 on a design formed as described in Method 1 from an orbit or a union of orbits, and the codes are the codes of the associated 1-design. 1. For J2 of degree 100, J¯2 is the full automorphism group of the design with parameters 1-(100, 36, 36), and it is the automorphism group of the self-orthogonal doubly-even [100, 36, 16]2 binary code of this design. 2. For J2 of degree 280, J¯2 is the full automorphism group of the design with parameters 1-(280, 108, 108), and it is the automorphism group of the self-orthogonal doubly-even [280, 14, 108]2 binary code of this design. The weight distribution of this code is , , , , , , , ,

Thus the words of minimum weight (i.e. 108) are the incidence vectors of the design. 3. For J2 of degree 315, J¯2 is the full automorphism group of the design with parameters 1-(315, 64, 64) (by taking the union of the two orbits of length 32), and it is the automorphism group of the self orthogonal doubly-even [315, 28, 64]2 binary code of this design. The weight distribution of the code is as follows: ,,,,, ,,,, ,,,, ,,, ,,,, ,,,

Thus the words of minimum weight (i.e. 64) are the incidence vectors of the blocks of the design. Furthermore, the designs from the two orbits of length 32 in this case, i.e. 1-(315, 32, 32) designs, each have J2 as their automorphism group. Their binary codes are equal, and are [315, 188]2 codes, with hull the 28dimensional code described above. The automorphism group of this 188dimensional code is again J¯2 . The minimum weight is at most 32. This is also the binary code of the design from the orbit of length 160. 4. For J2 of degree 315, J¯2 is the full automorphism group of the design with parameters 1-(315, 160, 160) and it is the automorphism group of the [315, 265]5 5-ary code of this design. This code is also the 5-ary code of the design obtained from the orbit of length 10, and from that of the orbit of length 80, so we can deduce that the minimum weight is at most 10. The hull is a [315, 15, 155]5 code and again with J¯2 as full automorphism group. 5. For J2 of degree 315, J¯2 is the full automorphism group of the design with parameters 1-(315, 80, 80) from the orbit of length 80, and it is the automorphism group of the self-orthogonal doubly-even [315, 36, 80]2 binary code of this design. The minimum words of this code are precisely the 315 incidence vectors of the blocks of the design.

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In [19] we used the construction described in Method 1 to obtain all irreducible modules of J1 (as codes) over the prime fields F2 , F3 , F5 . We also showed that most of those of J2 can be represented in this way as the code, the dual code or the hull of the code of a design, or of codimension 1 in one of these. For J2 , if no such code was found for a particular irreducible module, then we checked that it could not be so represented for the relevant degrees of the primitive permutation representations up to and including 1008. In summary, we obtained: Proposition 10 Using the construction described in Method 1 above (see Theorem 5 and Lemma 6), taking unions of orbits, the following constructions of the irreducible modules of the Janko groups J1 and J2 as the code, the dual code or the hull of the code of a design, or of codimension 1 in one of these, over Fp where p = 2, 3, 5, were found to be possible: 1. J1 : all the seven irreducible modules for p = 2, 3, 5; 2. J2 : all for p = 2 apart from dimensions 12, 128; all for p = 3 apart from dimensions 26, 42, 114, 378; all for p = 5 apart from dimensions 21, 70, 189, 300. For these exclusions, none exist of degree ≤ 1008. Note: 1. We do not claim that we have all the constructions of the modular representations as codes; we were seeking mainly existence. We give below three self-orthogonal binary codes of dimension 20 invariant under J1 of lengths 1045, 1463, and 1540. These are irreducible by [16] or Magma data. In all cases the Magma simgps library is used for J1 and J2 . 1. J1 of Degree 1045 [1045, 20, 456]2 code; dual code: [1045, 1025, 4]2 \\Orbit lengths of stabilizer of a point: [ 1, 8, 28, 56, 56, 56, 168, 168, 168, 168, 168 ]; \\Orbits chosen: ##1,3,5,10,11 \\Defining block is the union of these, length 421 1-(1045, 421, 421) Design with 1045 blocks \\C is the code of the design, of dimension 21 \\The 20-dimensional code is Ch:= C meet Dual(C) =Hull(C) > WeightDistribution(Ch); [ , , , , , , , , , , , , , ]. Those of weight 456, 504, 544, 552, 624, 608 are single orbits; the others split. >WeightDistribution(C); [ , , , , , , , < 496, 87780>, , , , , , , , , , , , , , , , , , , , ].

2. J1 of Degree 1463 [1463, 20, 608]2 code; dual code: [1463, 1443, 3]2

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\\Orbit lengths of stabilizer of a point: [ 1, 12, 15, 15, 20, 20, 60, 60, 60, 60, 60, 60, 60, 60, 60, 120, 120, 120, 120, 120, 120, 120 ] \\Orbits chosen ##18,21 \\Defining block is union of these, of length 240 1-(1463, 240, 240) Design with 1463 blocks \\C is the code of the design, of dimension 492 \\The 20-dimensional code is Ch:= C meet Dual(C) =Hull(C) WD(Ch); [ , , , , , , , , , , , , , ]

3. J1 of Degree 1540 [1540, 20, 640]2 code; dual code: [1540, 1520, 4]2 \\Orbit lengths of stabilizer of a point: [ 1, 19, 38, 38, 38, 38, 57, 57, 57, 57, 57, 57, 114, 114, 114, 114, 114, 114, 114, 114, 114 ] \\Orbits chosen ##7,13 \\Defining block is the union of these, length 171 1-(1540, 171, 171) Design with 1540 blocks \\C is the code of the design, of dimension 592 \\The code of dimension 20 is Ch:=C meet Dual(C) WD(Ch); [ , , , , , , , , , ]

We now look at the smallest representations for J2 . We have not been able to find any of dimension 12, and none can exist for degree ≤ 1008, as we have verified computationally by examining the permutation modules. We give below four representations of J2 acting on self-orthogonal binary codes of small degree that are irreducible or indecomposable codes over J2 . The full automorphism group of each of these codes is J¯2 . 1. J2 of Degree 100, dimension 36 [100, 36, 16]2 code; dual code: [100, 64, 8]2 \\Orbit lengths of stabilizer of a point: [1, 36, 63] 1-(100, 36, 36) Design with 100 blocks \\ Orbit #2 gave a block of the design [ , , , , , , , , , , , , , , , , ]

This code C = C36 of dimension 36 is irreducible, by Magma. The dual code C64 = C ⊥ has an invariant subcode C63 of dimension 63 that is spanned by the weight-8 vectors and that contains j and C36 . All these codes are indecomposable, by Magma. The full automorphism group of this code is J¯2 .

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2. J2 of Degree 280, dimension 13 [280, 13, 128]2 code; dual code: [280, 267, 4]2 \\Orbit lengths of stabilizer of a point: [1, 36, 108, 135] \\Orbit #3 gave a block of the design 1-(280,108,108) Design with 280 blocks \\Weight distribution of its 14-dimensional binary code [ , , , , , , , , ] Dual code: [280,266,4] \\Weight distribution of reducible but indecomposable 13-dimensional code [ , , , , , ]

This code has the invariant subcode of dimension 1 generated by the allone vector, so it is reducible. However, we checked the orbits of all the other words and found that there are no other invariant subcodes. It is thus indecomposable. The full automorphism group of these codes is J¯2 . 3. J2 of Degree 315, dimension 28 [315, 28, 64]2 code; dual code: [315, 287, 3]2 \\Orbit lengths of stabilizer of a point: [ 1, 10, 32, 32, 80, 160 ] \\Orbits ## 3 and 4 chosen 1-(315, 64, 64) Design with 315 blocks \\Weight distribution of its 28-dimensional binary code [ , , , , , , , , , , , , , , , , , , , , , , , ]

The code is an irreducible module over J2 , by Magma. The full automorphism group of this code is J¯2 . 4. J2 of Degree 315, dimension 36 [315, 36, 80]2 code; dual code: [315, 279, 5]2 \\Orbit lengths of stabilizer of a point: [ 1, 10, 32, 32, 80, 160 ] \\chose the orbit of length 80 1-(315, 80, 80) Design with 315 blocks 36 =Dim(C) dim hull 36 //Weight distribution of the 36-dimensional code [ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ]

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The code is an irreducible module over J2 , by Magma. The full automorphism group of this code is J¯2 . For F one of the fields Fp for p = 2, 3, 5 and n the degree of the permutation representation, in [19] we demonstrated some cases where the full space F n can be completely decomposed into G-modules, where G = J1 , J2 , using codes obtained by our construction. In all cases Cm denotes an indecomposable linear code of dimension m over the relevant field and group. If the codes were irreducible they were obtained according to our method and were listed in [19]. For example • For J1 of degree 1045 over F2 , the full space can be completely decomposed into J1 -modules, that is: = C76 ⊕ C112 ⊕ C360 ⊕ C496 ⊕ F2 j, F1045 2 where all but C496 are irreducible. C496 has composition factors of dimensions 20, 112, 1, 76, 20, 1, 112, 20, 1, 1, 112, 20. Also ) = F2 j ⊕ C20 ⊕ C76 ⊕ C112 ⊕ C360 , S = Socle(F1045 2 with dim(S) = 569. • For J2 of degree 315 over F2 we have: F2315 = C160 ⊕ C154 ⊕ F2 j, ⊥ is the binary code of the where C160 is irreducible and C154 ⊕ F2 j = C160 and F280 are 1-(315, 33, 33) design from orbits #1 and #4. (Note that F100 2 2 indecomposable as J2 modules.) • For J2 of degree 100 over F3 we have:

F3100 = C36 ⊕ C63 ⊕ F3 j. • For J2 of degree 280 over F3 we have: F3280 = C63 ⊕ C216 ⊕ F3 j, where C216 is the code of the 1-(280, 135, 135) design obtained from the orbit # 4. • for J2 of degree 525 over F5 we have: F5525 = C175 ⊕ C100 ⊕ C250 , where C175 is irreducible and C100 is the dual of the code C of the 1(525, 140, 140) design obtained from the orbits #2, #3 , #4, and C250 = ⊥ . C ∩ C175

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4.1.2. The Conway group Co2 The Leech lattice is a certain 24-dimensional Z submodule of the Euclidean space R24 whose automorphism group is the double cover 2. Co1 of the Conway group Co1 . The Conway groups Co2 and Co3 are stabilizers of sublattices of the Leech lattice. The subgroup structure of Co2 is discussed in Wilson [36] and [35] using the following information. The group Co2 admits a 23-dimensional indecomposable representation over GF (2) obtained from the 24-dimensional Leech lattice by reducing modulo 2 and factoring out a fixed vector. The action of Co2 on the vectors of this 23-dimensional indecomposable GF (2) module (say M ) produces eight orbits, with stabilizers isomorphic to Co2 , U6 (2):2, 210 :M22 :2, M c L, HS:2, U4 (3).D8 , 21+8 + :S8 and M23 , respectively. The 23-dimensional indecomposable GF (2) module M contains an irreducible GF (2)-submodule N of dimension 22. We use TABLE III(a) given by Wilson in [35] to produce Table 5, which gives the orbit lengths and stabilizers for the actions of Co2 on M and N respectively. M -Stabilizer Co2 U6 (2) : 2

M -Orbit length 1 2300

M cL

47104

210 :M22 :2

46575

HS:2

476928

U4 (3).D8

1619200

M23

4147200

21+8 + :S8

2049300

N -Stabilizer

N -Orbit length

Co2 U6 (2) : 2

210 :M22 :2 HS:2

1 2300

46575 476928

U4 (3).D8

1619200

21+8 + :S8

2049300

Table 5. Action of Co2 on M and N

On the other hand, reduction modulo 2 of the 23-dimensional ordinary irreducible representation results in a decomposable 23-dimensional GF (2) representation. In [36] Wilson showed that Co2 has exactly eleven conjugacy classes of maximal subgroups. One of these subgroups is the group U6 (2):2 of index 2300. In Proposition 11, using this maximal subgroup, we construct the decomposable 23-dimensional GF (2)-representation as the binary code C892 of dimension 23 invariant under the action of Co2 . The action of Co2 on C892 produces 12 orbits with stabilizers isomorphic to Co2 (2 copies), U6 (2):2 (2 copies), 210 :M22 :2 (2 : S8 (2 copies) respectively. copies), HS:2 (2 copies), U4 (3).D8 (2 copies), 21+8 + Furthermore, C892 contains a binary code C1408 of dimension 22 invariant and irreducible under the action of Co2 . Notice that the 2-modular character table of Co2 is completely known (see [31]) and follows from it that the irreducible 22-dimensional GF (2) representation is unique and 22 is the smallest dimension for any non-trivial irreducible GF (2) module. Here we examine some designs Di and associated binary codes Ci constructed from a primitive permutation representation of degree 2300 of the sporadic simple group Co2 . For the full detail the readers are encouraged to see [27].

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We used Method 1 and constructed self-dual symmetric 1-designs Di and binary codes Ci , where i is an element of the set {891, 892, 1408, 1409, 2299}, from the rank-3 primitive permutation representation of degree 2300 of the sporadic simple group Co2 of Conway. The stabilizer of a point α in this representation is a maximal subgroup isomorphic to U6 (2):2, producing orbits {α}, Δ1 , Δ2 of lengths 1, 891 and 1408 respectively. The self-dual symmetric 1-designs Di are constructed from the sets Δ1 , {α} ∪ Δ1 , Δ2 , {α} ∪ Δ2 , and Δ1 ∪ Δ2 , respectively. We let Ω = {α} ∪ Δ1 ∪ Δ2 . We proved the following result: Proposition 11 Let G be the Conway group Co2 and Di and Ci where i is in the set {891, 892, 1408, 1409, 2299} be the designs and binary codes constructed from the primitive rank-3 permutation action of G on the cosets of U6 (2):2. Then the following holds: (i) Aut(D891 ) = Aut(D892 ) = Aut(D1408 ) = Aut(D1409 ) = Aut(C892 ) = Aut(C1408 ) = Co2 . (ii) dim(C892 ) = 23, dim(C1408 ) = 22, C892 ⊃ C1408 and Co2 acts irreducibly on C1408 . (iii) C891 = C1409 = C2299 = V2300 (GF (2)). (iv) Aut(D2299 ) = Aut(C891 ) = Aut(C1049 ) = Aut(C2299 ) = S2300 . The proof of the proposition follows from a series of lemmas. In fact we showed that the codes C892 and C1408 are of types [2300, 23, 892]2 and [2300, 22, 1024]2 respectively. Furthermore C892 = C1408 , j = C1408 ∪ {w + j : w ∈ C1408 } = C1408 ⊕ j , where j denotes the all-one vector. Let Wl denote the set of all codewords of C892 of weight l and let Al be the size of Wl . Then clearly Wl + {j} = W2300−l ⊂ C892 and |Wl | = Al = |W2300−l | = A2300−l . We found the weight distribution of C892 and then the weight distribution of C1408 follows. We also determined the structures of the stabilizers (Co2 )wl , for all nonzero weight l, where wl ∈ C1408 is a codeword of weight l. The structures of the stabilizers (Co2 )wl for C892 follows clearly from those of C1408 . We also showed that the code C1408 is the 22 dimensional irreducible representation of Co2 over GF (2) contained in the 23-dimensional decomposable C892 . It is also contained in the 23-dimensional indecomposable representation of Co2 over GF (2) discussed in ATLAS [5] and Wilson [35]. We should also mention that computation with Magma shows the codes over some other primes, in particular, p = 3 are of some interest. In a separate paper we plan to deal with the ternary codes invariant under Co2 [30].

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5. Method 2 Construction of 1-Designs and Codes from Maximal Subgroups and Conjugacy Classes of Elements: In this section we assume G is a finite simple group, M is a maximal subgroup of G, nX is a conjugacy class of elements of order n in G and g ∈ nX. Thus Cg = [g] = nX and |nX| = |G : CG (g)|. As in Section 3 let χM = χ(G|M ) be the permutation character afforded by the action of G on Ω, the set of all conjugates of M in G. Clearly if g is not conjugate to any element in M , then χM (g) = 0. The construction of our 1-designs is based on the following theorem. Theorem 12 Let G be a finite simple group, M a maximal subgroup of G and nX a conjugacy class of elements of order n in G such that M ∩ nX = ∅. Let B = {(M ∩nX)y |y ∈ G} and P = nX. Then we have a 1−(|nX|, |M ∩nX|, χM (g)) design D, where g ∈ nX. The group G acts as an automorphism group on D, primitive on blocks and transitive (not necessarily primitive) on points of D. Proof: First note that B = {M y ∩ nX|y ∈ G}. We claim that M y ∩ nX = M ∩ nX if and only if y ∈ M or nX = {1G }. Clearly if y ∈ M or nX = {1G }, then M y ∩ nX = M ∩ nX. Conversely suppose there exits y ∈ / M such that M y ∩ nX = M ∩ nX. Then maximality of M in G implies that G =< M, y > and hence M z ∩ nX = M ∩ nX for all z ∈ G. We can deduce that nX ⊆ M and hence < nX >≤ M. Since < nX > is a normal subgroup of G and G is simple, we must have < nX >= {1G }. Note that maximality of M and the fact < nX >≤ M , excludes the case < nX >= G. From above we deduce that b = |B| = |Ω| = [G : M ]. If B ∈ B, then k = |B| = |M ∩ nX| =

k  i=1

|[xi ]M | = |M |

k  i=1

1 , |CM (xi )|

where x1 , x2 , ..., xk are the representatives of the conjugacy classes of M that fuse to g. Let v = |P| = |nX| = [G : CG (g)]. Form the design D = (P, B, I), with point set P, block set B and incidence I given by xIB if and only if x ∈ B. Since the number of blocks containing an element x in P is λ = χM (x) = χM (g), we have produced a 1 − (v, k, λ) design D, where v = |nX|, k = |M ∩ nX| and λ = χm (g). The action of G on blocks arises from the action of G on Ω and hence the maximality of M in G implies the primitivity. The action of G on nX, that is on points, is equivalent to the action of G on the cosets of CG (g). So the action on points is primitive if and only if CG (g) is a maximal subgroup of G. 

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Remark 4 Since in a 1 − (v, k, λ) design D we have kb = λv, we deduce that k = |M ∩ nX| =

χM (g) × |nX| . [G : M ]

˜ design, where λ ˜ = ˜ the complement of D, is 1 − (v, v − k, λ) Also note that D, v−k λ× k . Remark 5 If λ = 1, then D is a 1 − (|nX|, k, 1) design. Since nX is the disjoint union of b blocks each of size k, we have Aut(D) = Sk ≀ Sb = (Sk )b : Sb . Clearly In this case for all p, we have C = Cp (D) = [|nX|, b, k]p , with Aut(C) = Aut(D). Remark 6 The designs D constructed by using Theorem 12 are not symmetric in general. In fact D is symmetric if and only if b = |B| = v = |P| ⇔ [G : M ] = |nX| ⇔ [G : M ] = [G : CG (g)] ⇔ |M | = |CG (g)|. 5.1. Some 1-designs and Codes from A7 A7 has five conjugacy classes of maximal subgroups, which are listed in Table 6. It has also 9 conjugacy classes of elements, some of which are listed in Table 7. No.

Structure

Index

Order

Max[1]

A6

7

360

Max[2]

P SL2 (7)

15

168

Max[3]

P SL2 (7)

15

168

Max[4]

S5

21

120

Max[5]

(A4 × 3):2

35

72

Table 6. Maximal subgroups of A7

We apply the Theorem 12 to the above maximal subgroups and few conjugacy classes of elements of A7 to construct several non-symmetric 1- designs. The corresponding binary codes are also constructed. nX

|nX|

CG (g)

Maximal Centralizer

2A 3A

105

D8 : 3

No

70

A4 × 3 ∼ = (22 × 3): 3

3B

280

No

3×3

No

Table 7. Some of the conjugacy classes of A7

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5.1.1. G = A7 , M = A6 and nX = 3A Let G = A7 , M = A6 and nX = 3A. Then b = [G : M ] = 7, v = |3A| = 70, k = |M ∩ 3A| = 40. Also using the character table of A7 , we have χM = χ1 + χ2 = 1a + 6a and hence χM (g) = 1+3 = 4 = λ, where g ∈ 3A. We produce a non-symmetric 1−(70, 40, 4) design D. A7 acts primitively on the 7 blocks. Since CA7 (g) = A4 × 3 is not maximal in A7 (sits in the maximal subgroup (A4 × 3):2 with index two), A7 ˜ is a 1 − (70, 30, 3) acts imprimitively on the 70 points. The complement of D, D, design. Computations with MAGMA [4] shows that the full automorphism group of D is Aut(D) ∼ = 235 :S7 ∼ = 25 ≀ S 7 , with |Aut(D)| = 239 .32 .5.7. Construction using MAGMA shows that the binary code C of this design is a [70, 6, 32]2 code. The code C is self-orthogonal with the weight distribution < 0, 1 >, < 32, 35 >, < 40, 28 > . Our group A7 acts irreducibility on C. If Wi denote the set of all words in C of weight i, then C =< W32 >=< W40 >, so C is generated by its minimum-weight codewords. The full automorphism group of C is Aut(C) ∼ = 235 :S8 with |Aut(C)| = 242 .32 .5.7, and we note that Aut(C) ≥ Aut(D) and that Aut(D) is not a normal subgroup of Aut(C). Furthermore C ⊥ is a [70, 64, 2]2 code and its weight distribution has been determined. Since the blocks of D are of even size 40, we have that j meets evenly ¯ i denote the set of all codewords in C ⊥ every vector of C and hence j ∈ C ⊥ . If W ¯ ¯ ¯ 4 | = 14035 and of weight i, then |W2 | = 35,, |W3 | = 840, |W ¯ 3 >, dim(< W ¯ 2 >= 35, dim(< W ¯ 4 >= 63. C ⊥ =< W Let eij denote the 2-cycle (i, j) in S7 , where {i, j} = s(w ¯2 ) is the support of a ¯ 2 . Then eij (w ¯ 2 >= codeword w ¯2 ∈ W ¯2 ) = w ¯2 , and < eij |{i, j} = s(w¯2 ), w ¯2 ∈ W 35 2 . Using MAGMA we can easily show that V = F270 is decomposable into indecomposable G-modules of dimension 40 and 30. We also have dim(Soc(V ) = 21 and Soc(V ) =< j > ⊕C ⊕ C14 , where C is our 6-dimensional code and C14 is an irreducible code of dimension 14. The structures the stabilizers Aut(D)wl and Aut(C)wl , where l ∈ {32, 40} are listed in Table 8 and 9.

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|Wl |

Aut(D)wl

35

235 :(A4 × 3):2

40(1)

7

235 :S6

40(2)

21

235 :(S5 :2)

32

221

Table 8. Stabilizer of a word wl in Aut(D)

l 32 40

|Wl |

Aut(D)wl

35

235 :(S4 × S4 ):2

28

235 :(S6 × 2)

Table 9. Stabilizer of a word wl in Aut(C)

5.1.2. G = A7 , M = A6 and nX = 2A Let G = A7 , M = A6 and nX = 2A. Then b = [G : M ] = 7, v = |2A| = 105, k = |M ∩ 2A| = 45. Also using the character table of A7 , we have χM = χ1 + χ2 = 1a + 6a and hence χM (g) = 1+2 = 3 = λ, where g ∈ 2A. We produce a non-symmetric 1−(105, 45, 3) design D. A7 acts primitively on the 7 blocks. Since CA7 (g) = D8 : 3 is not maximal in A7 (sits in the maximal subgroup (A4 × 3):2 with index three), A7 ˜ is a 1−(105, 60, 4) acts imprimitively on the 105 points. The complement of D, D, design. The full automorphism group of D is Aut(D) ∼ = S3 35 :S7 ∼ = S3 5 ≀ S7 , with |Aut(D)| = 242 .337 .5.7. Construction using MAGMA shows that the binary code C of this design is a [105, 7, 45]2 code. The weight distribution of C is < 0, 1 >, < 45, 28 >, < 48, 35 >, < 57, 35 >, < 60, 28 >, < 105, 1 > . We also have that Hull(C) is a [105, 6, 48] code and has the following weight distribution: < 0, 1 >, < 48, 35 >, < 60, 28 > . Note that C = Hull(C)⊕ < j >, and that our group A7 acts irreducibility on Hull(C). Also note that this result together with the result obtained in 5.1.2 imply that the 6-dimensional irreducible representation of A7 over GF (2) could be represented by two non-isomorphic codes, namely [105, 6, 48]2 and [70, 6, 32]2 codes. We also have

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C =< W45 >=< W57 >, so C is generated by its minimum-weight codewords. The full automorphism group of C is Aut(C) = Aut(D) and its structure was given above in 5.2.1. Using MAGMA we can easily show that V = F2105 is decomposable into indecomposable G-modules of dimension 1, 14, 20 and 70 (the first three are irreducible). We also have dim(Soc(V ) = 55 and that Soc(V ) =< j > ⊕C14 ⊕ C14 ⊕ C20 ⊕ Hull(C), where C = Hull(C)⊕ < j > is our 7-dimensional code and C14 and C20 are irreducible codes of dimension 14 and 20 respectively. 5.1.3. G = A7 , M = S5 and nX = 2A: 1 − (105, 25, 5) Design Let G = A7 , M = S5 and nX = 2A. Then b = [G : M ] = 21, v = |2A| = 105, k = |M ∩ 2A| = 25. Note that both conjugacy classes of involutions of S5 fuses to 2A. Also using the character table of A7 , we have χM = χ1 + χ2 + χ5 = 1a + 6a + 14a and hence χM (g) = 1 + 2 + 2 = 5 = λ, where g ∈ 2A. We produce a non-symmetric 1 − (105, 25, 5) design D. A7 acts primitively on the 21 blocks. Since CA7 (g) = D8 :3 is not maximal in A7 (sits in the maximal subgroup (A4 × 3):2 with index ˜ is a three), A7 acts imprimitively on the 105 points. The complement of D, D, 1 − (105, 80, 16) design. 5.1.4. G = A7 , M = P SL2 (7) and nX = 2A: 1 − (105, 21, 3) Design Let G = A7 , M = P SL2 (7) and nX = 2A. Then b = [G : M ] = 15, v = |2A| = 105, k = |M ∩ 2A| = 21. Also using the character table of A7 , we have χM = χ1 + χ6 = 1a + 14b and hence χM (g) = 1+2 = 3 = λ, where g ∈ 2A. We produce a non-symmetric 1−(105, 21, 3) design D. A7 acts primitively on the 15 blocks. Since CA7 (g) = D8 : 3 is not maximal in A7 (sits in the maximal subgroup (A4 ×3):2 with index three), A7 acts ˜ is a 1 − (105, 84, 12) imprimitively on the 105 points. The complement of D, D, design. 5.1.5. G = A7 , M = P SL2 (7) and nX = 3B: 1 − (280, 56, 3) Design Let G = A7 , M = P SL2 (7) and nX = 3B. Then b = [G : M ] = 15, v = |3B| = 280, k = |M ∩ 2A| = 56. Also using the character table of A7 , we have χM = χ1 + χ6 = 1a + 14b and hence χM (g) = 1+2 = 3 = λ, where g ∈ 3B. We produce a non-symmetric 1−(280, 56, 3) design D. A7 acts primitively on the 15 blocks. Since CA7 (g) = 3×3 ∈ Syl3 (A7 ) is not maximal in A7 (sits in the maximal subgroups A6 and (A4 × 3):2 with indices 40 and 8 respectively), A7 acts imprimitively on the 280 points. The complement ˜ is a 1 − (280, 224, 12) design. of D, D,

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5.2. Design and codes from P SL2 (q) The main aim of this section to develop a general approach to G = P SL2 (q), where M is the maximal subgroup that is the stabilizer of a point in the natural action of degree q + 1 on the set Ω. This is fully discussed in Subsection 5.2.1. We start this section by applying the results discussed for Method 2, particularly the Theorem 12, to all maximal subgroups and conjugacy classes of elements of P SL2 (11) to construct 1- designs and their corresponding binary codes. These are itemized bellow after Tables 10 and 11. The group P SL2 (11) has order 660 = 22 × 3 × 5 × 11, it has four conjugacy classes of maximal subgroups, which are listed in the table 10. It has also eight conjugacy classes of elements which we list in Table 11. No.

Order

Index

Structure

Max[1]

55

12

F55 = 11 : 5

Max[2]

60

11

A5

Max[3]

60

11

A5

Max[4]

12

55

D12

Table 10. Maximal subgroups of P SL2 (11)

nX

|nX|

CG (g)

Maximal Centralizer

2A

55

D12

Yes

3A

110

Z6

No

5A

132

Z5

No

5B

132

Z5

No

6A

110

Z6

No

11A

60

Z11

No

11B

60

Z11

No

Table 11. Conjugacy classes of P SL2 (11)

Max[1] 5A: D = 1 − (132, 22, 2), b = 12; C = [132, 11, 22]2 , C ⊥ = [132, 121, 2]2 ; Aut(D) = Aut(C) = 266 : S12 . 5B: As for 5A. 11A: D = 1 − (60, 5, 1), b = 12; C = [60, 12, 5]2 , C ⊥ = [60, 48, 2]2 ; Aut(D) = Aut(C) = (S5 )12 : S12 . 11B: As for 11A. Max[2] 2A : D = 1 − (55, 15, 3), b = 11; C = [55, 11, 15]2 , C ⊥ = [55, 44, 4]2 ; Aut(D) = P SL2 (11), Aut(C) = P SL2 (11) : 2. 3A : D = 1 − (110, 20, 2), b = 11; C = [110, 10, 20]2 , C ⊥ = [110, 100, 2]2 ; Aut(D) = Aut(C) = 255 : S11 .

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5A : D = 1 − (132, 12, 1), b = 11; C = [132, 11, 12]2 , C ⊥ = [132, 121, 2]2 ; Aut(D) = Aut(C) = (S12 )11 : S11 . 5B : As for 5A. Max[3] As for Max[2]. Max[4] 2A : D = 1 − (55, 7, 7), b = 55; C = [55, 35, 4]2 , C ⊥ = [55, 20, 10]2 ; Aut(D) = Aut(C) = P SL2 (11) : 2. 3A : D = 1 − (110, 2, 1), b = 55; C = [110, 55, 2]2 , C ⊥ = [110, 55, 2]2 ; Aut(D) = Aut(C) = 255 : S55 . 6A : As for 3A. 5.2.1. G = P SL2 (q) of degree q + 1, M = G1 Let G = P SL2 (q), let M be the stabilizer of a point in the natural action of degree q + 1 on the set Ω. Let M = G1 . Then it is well known that G acts sharply 2-transitive on Ω and M = Fq : Fq∗ = Fq : Zq−1 , if q is even, and M = Fq : Z q−1 , 2 if q is odd. Since G acts 2-transitively on Ω, we have χ = 1 + ψ where χ is the permutation character of the action and ψ is an irreducible character of G of degree q. Also since the action is sharply 2-transitive, only 1G fixes 3 distinct elements of Ω. Hence for all 1G = g ∈ G we have λ = χ(g) ∈ {0, 1, 2}. Proposition 13 For G = P SL2 (q), let M be the stabilizer of a point in the natural action of degree q + 1 on the set Ω. Let M = G1 . Suppose g ∈ nX ⊆ G is an element fixing exactly one point, and without loss of generality, assume g ∈ M . Then the replication number for the associated design is r = λ = 1. We also have (i) If q is odd then |g G | = 21 (q 2 − 1), |M ∩ g G | = 21 (q − 1), and D is a 1-( 21 (q 2 − 1), 21 (q − 1), 1) design with q + 1 blocks and Aut(D) = S 21 (q−1) ≀ Sq+1 = (S 21 (q−1) )q+1 : Sq+1 . For all p, C = Cp (D) = [ 21 (q 2 −1), q +1, 21 (q −1)]p , with Aut(C) = Aut(D). (ii) If q is even then |g G | = (q 2 − 1), |M ∩ g G | = (q − 1), and D is a 1((q 2 − 1), (q − 1), 1) design with q + 1 blocks and Aut(D) = S(q−1) ≀ Sq+1 = (S(q−1) )q+1 : Sq+1 . For all p, C = Cp (D) = [(q 2 − 1), q + 1, q − 1)]p , with Aut(C) = Aut(D). Proof: Since χ(g) = 1, we deduce that ψ(g) = 0. We now use the character table and conjugacy classes of P SL2 (q) (for example see [12]): (i) For q odd, there are two types of conjugacy classes with ψ(g) = 0. In both cases we have |CG (g)| = q and hence |nX| = |g G | = |P SL2 (q)|/q = (q 2 − 1)/2. Since b = [G : M ] = q + 1 and k=

1 × (q 2 − 1)/2 χ(g) × |nX| = = (q − 1)/2, [G : M ] q+1

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the results follow from Remark 5. class with (ii) For q even, P SL2 (q) = SL2 (q) and there is only one conjugacy

10 ψ(g) = 0. A class representative is the matrix g = with |CG (g)| = q 11 and hence |nX| = |g G | = |P SL2 (q)|/q = (q 2 − 1). Since b = [G : M ] = q + 1 and k=

1 × (q 2 − 1) χ(g) × |nX| = = q − 1, [G : M ] q+1

the results follow from Remark 5.  If we have λ = r = 2 then a graph (possibly with multiple edges) can be defined on b vertices, where b is the number of blocks, i.e. the index of M in G, by stipulating that the vertices labelled by the blocks bi and bj are adjacent if bi and bj meet. Then the incidence matrix for the design is an incidence matrix for the graph. In the case where the graph is an undirected graph without multiple edges the following result from [9, Lemma] can be used. Lemma 14 ([9]) Let Γ = (V, E) be a regular graph with |V | = N , |E| = e and valency v. Let G be the 1-(e, v, 2) incidence design from an incidence matrix A for Γ. Then Aut(Γ) = Aut(G). Note: If the graph Γ is also connected, then it is an easy induction to show that rankp (A) ≥ |V | − 1 for all p with obvious equality when p = 2. If in addition (as happens for some classes of graphs, see [9,24,23]) the minimum weight is the valency and the words of this weight are the scalar multiples of the rows of the incidence matrix, then we also have Aut(Cp (G)) = Aut(G). Proposition 15 For G = P SL2 (q), let M be the stabilizer of a point in the natural action of degree q + 1 on the set Ω. Let M = G1 . Suppose g ∈ nX ⊆ G is an element fixing exactly two points, and without loss of generality, assume g ∈ M = G1 and that g ∈ G2 . Then the replication number for the associated design is r = λ = 2. We also have (i) If g is an involution, so that q ≡ 1 (mod 4), the design D is a 1-( 21 q(q + 1), q, 2) design with q + 1 blocks and Aut(D) = Sq+1 . Furthermore C2 (D) = [ 21 q(q + 1), q, q]2 , Cp (D) = [ 21 q(q + 1), q + 1, q]p if p is an odd prime, and Aut(Cp (D)) = Aut(D) = Sq+1 for all p. (ii) If g is not an involution, the design D is a 1-(q(q + 1), 2q, 2) design 1 with q + 1 blocks and Aut(D) = 2 2 q(q+1) : Sq+1 . Furthermore C2 (D) = [q(q + 1), q, 2q]2 , Cp (D) = [q(q + 1), q + 1, 2q]p if p is an odd prime, and 1 Aut(Cp (D)) = Aut(D) = 2 2 q(q+1) : Sq+1 for all p. Proof: A block of the design constructed will be M ∩ g G . Notice that from elementary considerations or using group characters we have that the only powers of g that are conjugate to g in G are g and g −1 . Since M is transitive on Ω \ {1}, g M and (g −1 )M give 2q elements in M ∩ g G if o(g) = 2, and q if o(g) = 2.

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These are all the elements in M ∩ g G since Mj is cyclic so if h1 , h2 ∈ Mj and h1 = g x1 , h2 = g x2 for some x1 , x2 ∈ G, then h1 is a power of h2 , so they can only be equal or inverses of one another. (i) In this case by the above k = |M ∩ g G | = q and hence |nX| =

q × (q + 1) k × [G : M ] = . χ(g) 2

So D is a 1-( 21 q(q + 1), q, 2) design with q + 1 blocks. An incidence matrix of the design is an incidence matrix of a graph on q + 1 points labelled by the rows of the matrix, with the vertices corresponding to rows ri and rj being adjacent if there is a conjugate of g that fixes both i and j, giving an edge [i, j]. Since G is 2-transitive, the graph we obtain is the complete graph Kq+1 . The automorphism group of the design is the same as that of the graph (see [9]), which is Sq+1 . By [23], C2 (D) = [ 21 q(q + 1), q, q]2 and Cp (D) = [ 21 q(q +1), q +1, q]p if p is an odd prime. Further, the words of the minimum weight q are the scalar multiples of the rows of the incidence matrix, so Aut(Cp (D)) = Aut(D) = Sq+1 for all p. (ii) If g is not an involution, then k = |M ∩ g G | = 2q and hence |nX| =

2q × (q + 1) k × [G : M ] = = q(q + 1). χ(g) 2

So D is a 1-(q(q + 1), 2q, 2) design with q + 1 blocks. In the same way we define a graph from the rows of the incidence matrix, but in this case we have the complete directed graph. 1 The automorphism group of the graph and of the design is 2 2 q(q+1) : Sq+1 . Similarly to the previous case, C2 (D) = [q(q + 1), q, 2q]2 and Cp (D) = [q(q +1), q +1, 2q]p if p is an odd prime. Further, the words of the minimum weight 2q are the scalar multiples of the rows of the incidence matrix, so 1 Aut(Cp (D)) = Aut(D) = 2 2 q(q+1) : Sq+1 for all p.  We end this subsection by giving few examples of designs and codes constructed, using Propositions 13 and 15, from P SL2 (q) for q ∈ {16, 17, 19}, where M is the stabilizer of a point in the natural action of degree q +1 and g ∈ nX ⊆ G is an element fixing exactly one or two points. Example 1 (P SL2 (16)) 1. g is an involution having cycle type 11 28 , r = λ = 1: D is a 1 − (255, 15, 1) design with 17 blocks. For all p, C = Cp (D) = [255, 17, 15]p , with Aut(C) = Aut(D) = S15 ≀ S17 = (S15 )17 : S17 . 2. g is an element of order 3 having cycle type 12 35 , r = λ = 2: D is a 1 − (272, 32, 2) design with 17 blocks. C2 (D) = [272, 16, 32]2 and Cp (D) = [272, 17, 32]p for odd p. Also for all p we have Aut(Cp (D)) = Aut(D) = 2136 : S17 . Example 2 (P SL2 (17))

Note that 17 ≡ 1 (mod 4).

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1. g is an element of order 17 having cycle type 11 171 , r = λ = 1: D is a 1 − (144, 8, 1) design with 18 blocks. For all p, C = Cp (D) = [144, 18, 8]p , with Aut(C) = Aut(D) = S8 ≀ S18 = (S8 )18 : S18 . 2. g is an involution having cycle type 12 28 , r = λ = 2: D is a 1 − (153, 17, 2) design with 18 blocks. C2 (D) = [153, 17, 17]2 and Cp (D) = [153, 18, 17]p for odd p. Also for all p we have Aut(Cp (D)) = Aut(D) = S18 . 3. g is an element of order 4 having cycle type 12 44 , r = λ = 2: D is a 1 − (306, 34, 2) design with 18 blocks. C2 (D) = [306, 17, 34]2 and Cp (D) = [306, 18, 34]p for odd p. Also for all p we have Aut(Cp (D)) = Aut(D) = 2153 : S18 . 4. g is an element of order 8 having cycle type 12 82 , r = λ = 2: D is a 1 − (306, 34, 2) design with 18 blocks. C2 (D) = [306, 17, 34]2 and Cp (D) = [306, 18, 34]p for odd p. Also for all p we have Aut(Cp (D)) = Aut(D) = 2153 : S18 . Example 3 (P SL2 (19)) 1. g is an element of order 19 having cycle type 11 191 , r = λ = 1: D is a 1 − (180, 9, 1) design with 20 blocks. For all p, C = Cp (D) = [180, 20, 9]p , with Aut(C) = Aut(D) = S9 ≀ S20 = (S9 )20 : S20 . 2. g is an element of order 3 having cycle type 12 36 , r = λ = 2: D is a 1 − (380, 38, 2) design with 20 blocks. C2 (D) = [360, 19, 38]2 and Cp (D) = [360, 20, 38]p for odd p. Also for all p we have Aut(Cp (D)) = Aut(D) = 2190 : S20 . 5.3. Some 1-designs from the Janko group J1 The Janko group J1 of order 23 × 3 × 5 × 7 × 11 × 19 has seven conjugacy classes of maximal subgroups, which were listed in the table 1. It has also 15 conjugacy classes of elements some of which are listed in Table 12. nX

|nX|

2A

1463

3A

5852

CG (g)

Maximal Centralizer

2 × A5

Yes

D6 × 5

No

Table 12. Some of the conjugacy classes of J1

We apply the Theorem 12 to the maximal subgroups and few conjugacy classes of elements of J1 to construct several 1- designs. 5.3.1. G = J1 , M = P SL2 (11) and nX = 2A: 1 − (1463, 55, 10) Design Let G = J1 , M = P SL2 (11) and nX = 2A. Then b = [G : M ] = 266, v = |2A| = 1463, k = |M ∩ 2A| = 55. Also using the character table of J1 , we have χM = χ1 + χ2 + χ4 + χ6 = 1a + 56a + 56b + 76a + 77a

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and hence χM (g) = 1 + 0 + 0 + 4 + 5 = 10 = λ, where g ∈ 2A. We produce a non-symmetric 1−(1463, 55, 10) design D. Since CG (g) = 2×A5 is also a maximal subgroup of J1 , J1 acts primitively on blocks and points. The complement of D, ˜ is a 1 − (1463, 1408, 256) design. D, 5.3.2. G = J1 , M = 2 × A5 and nX = 2A: 1 − (1463, 31, 31) Design Let G = J1 , M = 2 × A5 and nX = 2A. Then b = [G : M ] = 1463, v = |2A| = 1463. It is easy to see that M = 2 × A5 has three conjugacy classes of order 2, namely x1 = z, x2 = α and x3 = zα, that fuse to 2A with corresponding centralizer orders 120, 8 and 8. Now by using Corollary 3 we have

λ = χM (g) =

3  120 120 120 |CG (g)| = + + = 31, |C (x )| 120 8 8 M i i=1

where g ∈ 2A. Alternatively we can use the character table of J1 to find that χM = χ1 + χ2 + χ3 + 2χ4 + 2χ6 + χ9 + χ10 + χ11 + 2χ12 + 2χ15 , and χM (g) = 1 + 0 + 0 + 8 + 10 + 0 + 0 + 0 + 10 + 2 = 31 = λ. In this case clearly k = |M ∩ 2A| = λ = 31, and we produce a symmetric 1 − (1463, 31, 31) design D. Obviously J1 acts primitively on blocks and points. ˜ is a 1 − (1463, 1432, 1432) design. The complement of D, D, 5.3.3. G = J1 , M = P SL2 (11) and nX = 3A: 1 − (5852, 110, 5) Design Let G = J1 , M = P SL2 (11) and nX = 3A. Then b = [G : M ] = 266, v = |3A| = 5852, k = |M ∩ 3A| = 110. Also using the character table of J1 , we have χM = χ1 + χ2 + χ4 + χ6 = 1a + 56a + 56b + 76a + 77a and hence χM (g) = 1 + 4 + 1 − 1 = 5 = λ, where g ∈ 3A. We produce a nonsymmetric 1 − (5852, 110, 5) design D. Since CG (g) = D6 × 5 is not a maximal subgroup of J1 , J1 acts primitively on 266 blocks but imprimitively on 5852 points. ˜ is a 1 − (5852, 5742, 261) design. The complement of D, D,

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5.3.4. G = J1 , M = P SL2 (11) and nX = 3A: 1 − (5852, 20, 5) Design Let G = J1 , M = 2 × A5 and nX = 3A. Then b = [G : M ] = 1463, v = |3A| = 5852, k = |M ∩ 3A| = 20. It is easy to see that M = 2 × A5 has only one conjugacy class of elements of order 3, which fuses to 3A, with the corresponding centralizer order 6. Now by using Corollary 3 we have λ = χM (g) =

30 |CG (g)| = = 5, |CM (x)| 6

where g ∈ 3A. Alternatively we can use the character χM as in Subsection 5.3.2 to find that

χM (g) = 1 + 2 + 2 + 2 − 2 + 0 + 0 + 0 + 2 − 2 = 5 = λ, where g ∈ 3A. We produce a non-symmetric 1 − (5852, 20, 5) design D. Since CG (g) = D6 × 5 is not a maximal subgroup of J1 , J1 acts primitively on the ˜ is a 1463 blocks but imprimitively on the 5852 points. The complement of D, D, 1 − (5852, 5832, 1458) design.

References [1] F. Ali, Fischer-Clifford Theory for Split and non-Split Group Extensions, PhD Thesis, University of Natal, 2001. [2] E. F. Assmus, Jr. and J. D. Key, Designs and their Codes, Cambridge University Press, 1992 (Cambridge Tracts in Mathematics, Vol. 103, Second printing with corrections, 1993). [3] B. Bagchi, A regular two-graph admitting the Hall-Janko-Wales group, Combinatorial mathematics and applications (Calcutta, 1988), Sankhy¯ a, Ser. A 54 (1992), 35–45. [4] W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of Sydney, November 1994. [5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, An Atlas of Finite Groups, Oxford University Press, 1985. [6] A. E. Brouwer, Strongly regular graphs, in Charles J. Colbourn and Jeffrey H. Dinitz, editors, The CRC Handbook of Combinatorial Designs, pages 667–685. CRC Press, Boca Raton, 1996. VI.5. [7] L. Finkelstein, The maximal subgroups of Janko’s sinple group of order 50, 232, 960, J. Algebra, 30 (1974), 122–143. [8] L. Finkelestein and A. Rudvalis, Maximal subgroups of the Hall-Janko-Wales group, J. Algebra, 24 (1977), 486–493. [9] W. Fish, J. D. Key, and E. Mwambene, Codes from the incidence matrices and line graphs of Hamming graphs, submitted. [10] M. S. Ganief, 2-Generations of the Sporadic Simple Groups, PhD Thesis, University of Natal, 1997. [11] The GAP Group, GAP - Groups, Algorithms and Programming, Version 4.2 , Aachen, St Andrews, 2000, (http://www-gap.dcs.st-and.ac.uk/~gap). [12] K. E. Gehles, Ordinary characters of finite special linear groups, MSc Dissertaion, University of St Andrews, 2002.

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[13] W. Haemers, C. Parker, V. Pless, and V. D. Tonchev, A design and a code invariant under the simple group Co3, J. Combin. Theory, Ser. A, 62 (1993), 225-233. [14] W. C. Huffman, Codes and groups, in V. S. Pless and W. C. Huffman, editors, Handbook of Coding Theory, pages 1345–1440, Amsterdam: Elsevier, 1998, Volume 2, Part 2, Chapter 17. [15] I. M. Isaacs, Character Theory of Finite Groups, Academic Press, San Diego, 1976. [16] C. Jansen, K. Lux, R. Parker, and R. Wilson. An Atlas of Brauer Characters, Oxford Scientific Publications, Clarendon Press, 1995. LMS Monographs New Series 11. [17] J. D. Key and J. Moori, Designs, codes and graphs from the Janko groups J1 and J2 , J. Combin. Math. and Combin. Comput., 40 (2002), 143–159. [18] J. D. Key and J. Moori, Correction to: ”Codes, designs and graphs from the Janko groups J1 and J2 [J. Combin. Math. Combin. Comput., 40 (2002), 143–159], J. Combin. Math. Combin. Comput., 64 (2008), 153. [19] J. D. Key and J. Moori, Some irreducible codes invariant under the Janko group, J1 or J2 , submitted. [20] J. D. Key and J. Moori, Designs and codes from maximal subgroups and conjugacy classes of finite simple groups, submitted. [21] J. D. Key, J. Moori, and B. G. Rodrigues, On some designs and codes from primitive representations of some finite simple group, J. Combin. Math. and Combin. Comput., 45 (2003), 3–19. [22] J. D. Key, J. Moori, and B. G. Rodrigues, Some binary codes from symplectic geometry of odd characteristic, Utilitas Mathematica, 67 (2005), 121-128. [23] J. D. Key, J. Moori, and B. G. Rodrigues, Codes associated with triangular graphs, and permutation decoding, Int. J. Inform. and Coding Theory, to appear. [24] J. D. Key and B. G. Rodrigues, Codes associated with lattice graphs, and permutation decoding, submitted. [25] W. Knapp and P. Schmid, Codes with prescribed permutation group, J. Algebra, 67 (1980), 415–435. [26] J. Moori and B. G. Rodrigues, A self-orthogonal doubly even code invariant under the M c L : 2, J. Comb. Theory, Series A, 110 (2005), 53–69. [27] J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2 , J. of Algerbra, 316 (2007), 649–661. [28] J. Moori and B. G. Rodrigues, A self-orthogonal doubly-even code invariant under Mc L, Ars Combinatoria, 91 (2009), 321–332. [29] J. Moori and B. G. Rodrigues, Some designs and codes invariant under the Higman-Sims group, Utilitas Mathematica, to appear. [30] J. Moori and B. Rodrigues, Ternary codes invariant under the simple group Co2 , under prepararion. [31] J. M¨ uller and J. Rosenboom, Condensation of induced representations and an application: the 2-modular decomposition numbers of Co2 , Computational methods for representations of groups and algebras (Essen, 1997), 309–321, Progr. Math., 173, Birkhuser, Basel, 1999. [32] J. J. Rotman, An Introduction to the Theory of Groups, volume 148 of Graduate Text in Mathematics, Springer-Verlag, 1994. [33] I. A. Suleiman and R. A. Wilson, The 2-modular characters of Conway’s group Co2 , Math. Proc. Cambridge Philos. Soc. 116 (1994), 275–283. [34] V. D. Tonchev, Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs, IEEE Trans. Info. Theory, 43 (1997), 1021-1025. [35] R. A. Wilson, Vector stabilizers and subgroups of Leech lattice groups, J. Algebra, 127 (1989), 387–408. [36] R. A. Wilson, The maximal subgroups of Conway’s group Co2 , J. Algebra, 84 (1983), 107–114.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-231

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Designs, strongly regular graphs and codes constructed from some primitive groups ´ a ,Vedrana MIKULIC ´ CRNKOVIC ´ a and B.G. RODRIGUES b Dean CRNKOVIC a Department of Mathematics, University of Rijeka, Omladinska 14, 51000 Rijeka, Croatia b School of Mathematical Sciences, University of KwaZulu-Natal, Durban 4041, South Africa Abstract. Let G be a finite group acting primitively on the sets Ω1 and Ω2 . We describe a construction of 1-designs with block set Ω1 and block set Ω2 , having G as an automorphism group. Applying this construction method we obtain a unital 2(q 3 +1, q+1, 1), and a semi-symmetric (q 4 −q 3 +q 2 , q 2 −q, (1)) from the unitary group U3 (q), where q = 3, 4, 5, 7. From the unital and the semi-symmetric design we build a projective plane P G(2, q 2 ). Further, we describe other combinatorial structures constructed from these unitary groups and structures constructed from U4 (2), U4 (3) and L2 (49). We also construct self-orthogonal codes obtained from the row span over F2 or F3 of the incidence (resp. adjacency) matrices of mostly self-orthogonal designs (resp. strongly regular graphs) defined by the action of the simple unitary groups U3 (q) for q = 3, 4, 7 and U4 (q) for q = 2, 3 and the linear group L2 (49) on the conjugacy classes of some of their maximal subgroups. Some of the codes are optimal or near optimal for the given length and dimension. Keywords. design, strongly regular graph, code, primitive simple group

Introduction The study of finite groups prompts many questions about the groups and related structures. An interplay between primitive groups, combinatorial designs and graphs has been established by the by now standard construction given in [34], which was later corrected in [35]. In that paper codes are obtained from symmetric 1-designs admitting a primitive action of the group, and such that the point and the block stabilizers are conjugate. The designs obtained in this way have the automorphism group acting primitively on the points and on the blocks. In particular codes with interesting properties having finite simple groups acting have been found by a series of subsequent papers. Recently, in [14] a generalization of the construction outlined in [34,35] was described. This new construction allows for 1-designs which are not necessarily symmetric, with stabilizers of a point and a block not necessarily conjugate, although the group acts primitively on the points and on the blocks of the design. This paper collects results on designs, graphs, and codes constructed using this generalized construction.

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In fact the construction presented in this paper also generalizes the constructions of a symmetric 2-(126, 36, 14), 2-(36, 15, 6) and 2-(144, 66, 30) designs form primitive representations of the groups U3 (5) : Z2 (see [41]) , U3 (3) (see [30]) and M12 (see [57]) respectively. Similarly those given in [24] where the author describes all symmetric 2-designs that admit a primitive action of rank 3.

An incidence structure is an ordered triple D = (P, B, I) where P and B are nonempty disjoint sets and I ⊆ P × B. The elements of the set P are called points, the elements of the set B are called blocks and I is called an incidence relation. If |P| = |B| then the incidence structure is called symmetric. The complement of D is the structure ˜ = (P, B, I), ˜ where I˜ = P × B − I. The dual structure of D is Dt = (B, P, I t ), D where (B, p) ∈ I t if and only if (P, B) ∈ I. Thus the transpose of an incidence matrix for D is an incidence matrix for Dt and we say that D is self dual if it is isomorphic to its dual. The incidence matrix of an incidence structure is a b × v matrix [mij ] where b and v are the number of blocks and points respectively, such that mij = 1 if the point Pj and block xi are incident, and mij = 0 otherwise. An isomorphism from one incidence structure to another is a bijective mapping of points to points and blocks to blocks which preserves incidence. An isomorphism from an incidence structure D onto itself is called an automorphism of D. The set of all automorphisms forms a group called the full automorphism group of D and is denoted by Aut(D). A t−(v, k, λ) design is a finite incidence structure (P, B, I) satisfying the following requirements: 1. |P| = v, 2. every element of B is incident with exactly k elements of P, 3. every t elements of P are incident with exactly λ elements of B. A semi-symmetric (v, k, (λ)) design is a finite incidence structure with v points and v blocks satisfying: 1. every point (block) is incident with exactly k blocks (points), 2. every pair of points (blocks) are incident with 0 or λ blocks (points). A 2-(v, k, λ) design is called a block design. A 2-(v, k, λ) design is called quasisymmetric if the number of points in the intersection of any two blocks takes only two values. A symmetric 2 − (v, k, 1) design is called a projective plane. Let G = (V, E, I) be a finite incidence structure. G is a graph if each element of E is incident with exactly two elements of V. The elements of V are called vertices and the elements of E are called edges. Two vertices u and v are called adjacent or neighbors if they are incident with the same edge. The number of neighbors of a vertex v is called the degree of v. If all the vertices of the graph G have the same degree k, then G is called k-regular. Define a square {0, 1}−matrix A = (auv ) labeled with the vertices of G in such a way that auv = 1 if and only if the vertices u and v are adjacent. The matrix A is called the adjacency matrix of the graph G. A graph G is called a strongly regular graph with parameters (n, k, λ, µ), and denoted by SRG(n, k, λ, µ) if G is k−regular graph with n vertices and if any two adjacent vertices have λ common neighbors and any two non-adjacent vertices have µ common neighbors.

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Let x and y (x < y) be the two cardinalities of block intersections in a quasisymmetric design D. The block graph of D has as vertices the blocks of D and two vertices are adjacent if and only if they intersect in y points. The block graph of a quasisymmetric 2 − (v, k, λ) design is strongly regular. In a 2 − (v, k, 1) design which is not a projective plane two blocks intersect in 0 or 1 points, therefore the block graph of this design is strongly regular (see [10]). Let D be a symmetric (v, k, λ) design which possesses a symmetric incidence matrix M with 1 everywhere on the diagonal. Then the matrix M − I is an adjacency matrix of a strongly regular graph G with parameters (v, k − 1, λ − 2, λ) (see [10]) and Aut(G) ≤ Aut(D). The code CF of the design D over the finite field F is the space spanned by the incidence vectors of the blocks over F . If the point set of D is denoted by P and the block set by B, and if Q2 is any subset 3 of P, then we will denote the incidence vector of Q by v Q . Thus CF = v B | B ∈ B , and is a subspace of F P , the full vector space of functions from P to F . All our codes will be linear codes, i.e. subspaces of the ambient vector space. If a code C over a field of order q is of length n, dimension k, and minimum weight d, then we write [n, k, d]q to show this information. An [n, k, d] code is optimal if the d is the largest possible minimum weight for any [n, k]  code over the corresponding field. n The weight enumerator of C is defined as WC (x) = i=0 Ai xi , where Ai denotes the ⊥ number of codewords of weight i in C. The dual code C is the orthogonal complement under the standard inner product (, ), i.e. C ⊥ = {v ∈ F n |(v, c) = 0 for all c ∈ C}. A code C is self-orthogonal if C ⊆ C ⊥ and self-dual if C = C ⊥ . The hull of a design over the finite field F is the code obtained by intersecting both C and C ⊥ . The all-one vector will be denoted by 1, and is the constant vector of weight the length of the code. A binary code C is doubly-even if all codewords of C have weight divisible by four. Two linear codes of the same length and over the same field are equivalent if each can be obtained from the other by permuting the coordinate positions and multiplying each coordinate position by a non-zero field element. They are isomorphic if they can be obtained from one another by permuting the coordinate positions. An automorphism of a code is any permutation of the coordinate positions that maps codewords to codewords. For the structure of the simple groups we follow the ATLAS[12] notation.

1. The construction The following construction of symmetric 1-designs and regular graphs was described in [34, Proposition 1], used in [35] and later corrected in [36]: Theorem 1 Let G be a finite primitive permutation group acting on the set Ω of size n. Further, let α ∈ Ω, and let Δ = {α} be an orbit of the stabilizer Gα of α. If B = {Δg : g ∈ G} and, given δ ∈ Δ, E = {{α, δ}g : g ∈ G},

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then D = (Ω, B) is a symmetric 1 − (n, |Δ|, |Δ|) design. Further, if Δ is a self-paired orbit of Gα then Γ(Ω, E) is a regular connected graph of valency |Δ|, D is self-dual, and G acts as an automorphism group on each of these structures, primitive on vertices of the graph, and on points and blocks of the design. In [14] a generalization of the above construction is described; using this construction one obtains 1-designs which are not necessarily symmetric, and stabilizers of a point and a block that are not necessarily conjugate. For completeness we give the proof of this result and subsequent results. More details on these results can be found in [14]. Theorem 2 Let G be a finite permutation group acting*primitively on the sets Ω1 and Ω2 s of size m and n, respectively. Let α ∈ Ω1 and Δ2 = i=1 δi Gα , where δ1 , ..., δs ∈ Ω2 are representatives of distinct Gα -orbits. If Δ2 = Ω2 and B = {Δ2 g : g ∈ G}, s then (Ω2 , B) is a 1 − (n, |Δ2 |, i=1 |αGδi |) design with m blocks, and G acts as an automorphism group, primitive on points and blocks of the design. Proof: It is clear that the number of points v = n, since the point set is P = Ω2 , and also that each element of B consists of k = |Δ2 | elements of Ω2 . Since Δ2 is a Gα -orbit, we have Gα ⊆ G∆2 , where G∆2 is the setwise stabilizer of Δ2 . Since G is primitive on Ω1 , Gα is a maximal subgroup of G, and therefore G∆2 = Gα . The number of blocks is b = |Δ2 G| =

|G| |G| = |Ω1 | = m. = |G∆2 | |Gα |

Since G acts transitively on Ω1 and Ω2 the constructed structure is a 1-design, hence bk = vr, where each point is incident with r blocks. Therefore |Ω1 | |Δ2 | = |Ω2 | r, and consequently |G| |Gα | |G| = r. |Gα | |(Gα )δ | |Gδ | It follows that r=

|Gδ | |Gδ | = = |αGδ | = |Δ1 |. |(Gα )δ | |(Gδ )α |

 In the construction of the design D(G, α, δ) described in Theorem 2, instead of taking a single Gα -orbit, we can take Δ2 to be any union of Gα -orbits. In fact, this construction gives us all designs on which the group G acts primitively on points and blocks:

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Corollary 1 If the group G acts primitively on the points and the blocks of a 1-design D, then D can be obtained as described in Theorem 2, where Δ2 is a union of Gα -orbits. The set Δ1 of blocks incident with the point δ is a union of Gδ -orbits. Proof: Let α be any block of the design D. G acts transitively on the block set B of the design D, hence B = αG. Since G acts primitively on B, the stabilizer Gα is a maximal subgroup of G. Gα fixes α, so α is a union of Gα -orbits. In a similar way one concludes that Δ1 is a union of Gδ -orbits.  1.1. The incidence relation We can interpret the construction of a design from Theorem 2 in the following way: • the point set is Ω2 = δG, and the block set is Ω1 = αG, • the block αg ′ is incident with the set of points {δg : g ∈ Gα g ′ }. Let a point δg ∈ Ω2 be incident with a block αg ′ ∈ Ω1 . Then for g ∈ Gα g ′ there exists g ∈ Gα such that g = gg ′ . Hence we have ′

′

′

Gαg′ ∩ Gδg = Gαg′ ∩ Gδgg′ = Ggα ∩ Ggδg = (Gα ∩ Gδg )g =

′

−1

(Gα ∩ Ggδ )g = (Ggα

′

′

∩ Gδ )gg = (Gα ∩ Gδ )gg = (Gα ∩ Gδ )g .

If a point δg ∈ Ω2 is incident with the block α ∈ Ω1 , then Gα ∩ Gδg = (Gα ∩ Gδ )g . If the set {Gα ∩ Gδg | g ∈ G} contains Orb(Gα , Ω2 ) Gα -conjugacy classes, where Orb(Gα , Ω2 ) is the number of Gα -orbits on Ω2 , then each conjugacy class corresponds to one Gα -orbit, and the incidence relation in the design D(G, α, δ) can be defined as follows: • the block αg ′ is incident with the point δg if and only if Gαg′ ∩ Gδg is conjugate to Gα ∩ Gδ . Similarly, if the set {Gα ∩ Gδg | g ∈ G} contains Orb(Gα , Ω2 ) isomorphism classes, then the incidence in the design D(G, α, δ) can be defined as follows: • the block αg ′ is incident with the point δg if and only if Gαg′ ∩ Gδg ∼ = Gα ∩ G δ , 1.2. Conjugacy classes of simple groups Let G be a simple group and H1 and H2 be maximal subgroups of G. The conjugacy class of Hi , i = 1, 2, is denoted by cclG (Hi ) and |cclG (Hi )| = [G : NG (Hi )]. Denote gj the elements of cclG (Hi ), i = 1, 2, by Hig1 , H g2 , . . . , Hi i , ji = [G : NG (Hi )]. G acts primitively on cclG (H1 ) and cclG (H2 ) by conjugation. We can construct a primitive 1−design such that: • the point set of the design is cclG (H2 ), • the block set is cclG (H1 ), h h • the block H1gi is incident with the point H2 j if and only if H2 j ∩ H1gi ∼ = Gi , y x i = 1, . . . , k, where {G1 , ..., Gk } ⊂ {H2 ∩ H1 | x, y ∈ G}.

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Let us denote a 1−design constructed in this way by D(G, H2 , H1 ; G1 , ..., Gk ). From the conjugacy class of a maximal subgroup H of a simple group G one can construct a regular graph, in the following way: • the vertex set of the graph is cclG (H), • the vertex H gi is adjacent to the vertex H gj if and only if H gi ∩ H gj ∼ = Gi , i = 1, . . . , k, where {G1 , ..., Gk } ⊂ {H x ∩ H y | x, y ∈ G}. We denote this graph G(G, H; G1 , ..., Gk ). G acts primitively on the set of vertices of G(G, H; G1 , ..., Gk ). Remark 1 Let φ be an automorphism of a finite group G and H, H1 and H2 subgroups of G. Then • D(G, H2 , H1 ; G1 , ..., Gk ) ∼ = D(G, (H2 )φ, (H1 )φ; G1 , ..., Gk ), • G(G, H; G1 , ..., Gk ) ∼ = G(G, (H)φ; G1 , ..., Gk ). Remark 2 Described construction allows us to construct 1−design and regular graphs. In this paper we will consider only 1−designs that are 2−design and regular graphs that are strongly regular.

2. Results 2.1. Structures constructed from the unitary groups U3 (q), q = 3, 4, 5, 7 2.1.1. U3 (3) The unitary group U3 (3) is the simple group of order 6048. It possesses four maximal subgroups, up to conjugation (Table 1). Table 1. Maximal subgroups of the group U3 (3) (see [12]) No

Max. sub.

Order

Index

M1,1

(E9 : Z3 ) : Z8

216

28

M1,2

L2 (7)

168

36

M1,3

(Z4 × Z4 ) : S3 Z4 .S4

96 96

63 63

M1,4

Let G1 be a group isomorphic to the unitary group U3 (3) and M1,2 be a maximal subgroup of G1 isomorphic to L2 (7). The conjugacy class of the subgroup M1,2 in G1 is denoted by cclG1 (M1,2 ). Therefore, |cclG1 (H1 )| = |G1 : M1,2 | = 36. We denote the g1 g2 g36 elements of cclG1 (M1,2 ) by M1,2 , M1,2 , . . . , M1,2 . Using GAP ([43]), one can check that the intersection of two different elements gj gi of any kind M1,2 and M1,2 of the set cclG1 (M1,2 ) is either D8 or S4 . One can also gi check that for every element M1,2 , i = 1, . . . , 36, of the set cclG1 (M1,2 ), the cardinality gj gj ∼ gj gi |j = of the set {M1,2 | M1,2 ∩ M1,2 = S4 } is 14. Let us define sets Si = {M1,2 gj ∼ gi ∩ M1,2 i or M1,2 = S4 }, 1 ≤ i ≤ 36. For every 1 ≤ i, j ≤ 36, i = j, the set Si ∩ Sj has exactly 6 elements.

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This shows that the incidence structure D1 = D(G1 , M1,2 , M1,2 ; S4 , M1,2 ) is a symmetric 2-(36, 15, 6) design. The incidence matrix M1 of the design D1 is a symmetric matrix with 1 everywhere on the diagonal. Therefore, the design is self-dual and the matrix M1 − I is an adjacency matrix of a SRG(36, 14, 4, 6). An isomorphic strongly regular graph can be constructed directly from the group G1 , and it can be shown that the graph constructed is isomorphic to the graph G1 = G(G1 , M1,2 ; S4 ). The full automorphism group of the design D1 and the corresponding strongly regular graph has order 12096 and it is isomorphic to U3 (3) : Z2 ∼ = Aut(G1 ). The design D1 is isomorphic to the design described in [13] and in [24]. In a similar way, one can construct other block designs and strongly regular graphs from the group G1 . In the following tables (Table 2 and Table 3) we give all block design and strongly regular graph constructed from the G1 . Table 2. Block designs constructed from the group U3 (3) No

Structure

Parameters

Aut. group

D1

D(G1 , M1,2 , M1,2 ; S4 , M1,2 )

2-(36, 15, 6)

U3 (3) : Z2

D2 D3

D(G1 , M1,1 , M1,4 ; Z3 : Z8 ) D(G1 , M1,3 , M1,3 ; D16 YZ4 , Z2 × Z2 , M1,3 )

2-(28, 4, 1) 2-(63, 31, 15)

U3 (3) : Z2 L6 (2)

D4

D(G1 , M1,4 , M1,4 ; Z4 , Z4 × Z4 , M1,4 )

2-(63, 31, 15)

U3 (3) : Z2

D5

D(G1 , M1,1 , M1,3 ; Z8 )

2-(28, 12, 11)

S6 (2)

D6

D(G1 , M1,2 , M1,3 ; S3 )

2-(36, 16, 12)

S6 (2)

Table 3. Strongly regular graphs constructed from the group U3 (3) No

Structure

Parameters

Aut. group

G1

G(G1 , M1,2 ; S4 )

(36, 14, 4, 6)

U3 (3) : Z2

G2 G3

G(G1 , M1,3 ; D16 YZ4 , Z2 × Z2 ) G(G1 , M1,4 ; Z4 , Z4 × Z4 )

(63, 30, 13, 15) (63, 30, 13, 15)

S6 (2) U3 (3) : Z2

Notes on constructed structures: • The design D4 is isomorphic to the symmetric (63, 31, 15) design obtained from a generalized hexagon of order (2, 2) (see [25]). • The design D3 is the point-hyperplane design in the projective geometry P G(5, 2). • The design D5 is isomorphic to the derived design of the symplectic SDP design with parameters (64, 28, 12) (see [42]) and it is a quasi-symmetric SDP design. The block graph of this design is a SRG(63, 32, 16, 16) whose full automorphism group is isomorphic to S6 (2). That graph is the complement of the strongly regular graph G2 . • The design D2 is quasi-symmetric and its block graph is a SRG(63, 32, 16, 16) whose full automorphism group is isomorphic to U3 (3) : Z2 . That graph is the complement of the strongly regular graph G3 . • The design D6 is isomorphic to the residual design of the symplectic SDP design with parameters (64, 28, 12) (see [42]) and it is a quasi-symmetric SDP design, with block graph a SRG(63, 30, 13, 15) whose full automorphism group is isomorphic to S6 (2). That graph is isomorphic to the strongly regular graph G2 .

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The structure S1 = D(G1 , M1,4 , M1,4 ; Z4 × Z4 ) is a semi-symmetric design (63, 6, (1)) with the full automorphism group isomorphic to Aut(U3 (3)) ∼ = U3 (3) : Z2 . Let M1 and M be the incidence matrices of the constructed block design D2 and the semi-symmetric design S1 , respectively, and I28 be the identity matrix of order 28. Then the matrix ! I28 M1T P = M1 M is the incidence matrix of the Desarguesian projective plane P G(2, 9) with the full automorphism group isomorphic to the group P ΓL3 (9). P is a symmetric matrix, and therefore the projective plane admits a unitary polarity (for the definition see e.g. [23]) and the design D2 is the Hermitian unital in P G(2, 9). The absolute points and blocks are the G1 −conjugates of M1,1 , and the non-absolute points and blocks are the G1 −conjugates on M1,4 . 2.1.2. U3 (4) Let G2 be a group isomorphic to the unitary group U3 (4). Representatives of conjugacy classes of maximal subgroups are listed in the Table 4. Table 4. Maximal subgroups of the group U3 (4) No

Max. sub.

Order

M2,1

(E16 : E4 ) : Z15

960

65

M2,2 M2,3

A5 × Z5

300

208

E25 : S3 Z13 : Z3

150 39

416 1600

M2,4

Index

Using the describe method we constructed 3 block designs and 2 strongly regular graphs from the group G2 (Table 5 and Table6). Table 5. Block designs constructed from the group U3 (4) No

Structure

Parameters

Aut. group

D7 D8

D(G2 , M2,1 , M2,2 ; A4 × Z5 ) D(G2 , M2,1 , M2,3 ; Z10 )

2-(65, 5, 1) 2-(65, 15, 21)

U3 (4) : Z4 U3 (4) : Z4

D9

D(G2 , M2,1 , M2,4 ; Z3 )

2-(65, 26, 250)

U3 (4) : Z4

Table 6. Strongly regular graphs constructed from the group U3 (4) No

Structure

Parameters

Aut. group

G4

G(G2 , M2,2 ; E4 )

(208, 75, 30, 25)

U3 (4) : Z4

G5

G(G2 , M2,3 ; S3 )

(416, 100, 36, 20)

G2 (4) : Z2

The structure S2 = D(M2,2 , M2,2 ; E25 ) is a semi-symmetric design with parameters (208, 12, (1)) having U3 (4) : Z4 as full automorphism group. Let M1 and M be the incidence matrices of D7 and S2 respectively, and I65 be the identity matrix of order 65. Then the matrix

D. Crnković et al. / Designs, Strongly Regular Graphs and Codes

I65 M1

P =

M1T M

239

!

is the incidence matrix of a projective plane P G(2, 16), i.e., a symmetric (273, 17, 1) design. Since the matrix P is symmetric, the projective plane admits a unitary polarity. The absolute points and blocks are the conjugates of M1,1 , and the non-absolute points and blocks are the conjugates on M2,2 . The design D7 is the Hermitian unital in P G(2, 16). Notes on constructed structures: • The design D7 is resolvable (see [51]). • The design D7 is a quasi-symmetric whose block graph is a strongly regular graph with parameters (208, 75, 30, 25) and isomorphic to the graph G4 . • Each block of the design D8 is a union of three disjoint blocks of the design D7 which form a triangle in the projective plane P G(2, 16). A union of three disjoint blocks of D7 form a block of D8 if and only if a setwise stabilizer in Aut(D1 ) of the union is a group of order 600 isomorphic to M3,3 : Z4 . • Every block of D9 intersect 78 blocks of D7 in one point, 91 blocks in two points, and the remaining 39 blocks in four points. So, every block of D9 is a blocking set of the Hermitian unital D7 . • The group U3 (4) acts transitively on the graph G2 . The full automorphism group of the graph G5 is a group of order 503193600 isomorphic to G2 (4) : Z2 . This is the full automorphism group of the exceptional group G(2, 4), which is the simple group of order 251596800. Since the Janko group J2 is a subgroup of G(2, 4), J2 acts as an automorphism group of the graph G5 . The graph G5 was previously known. Namely, the Suzuki graph, a strongly regular graph with parameters (1782, 416, 100, 96), is locally G5 (see [49]). • The graph G5 can be constructed from the design D8 . Any two blocks of D8 intersect in 2,3, or 5 points. The graph which has as its vertices the blocks of D8 , two vertices being adjacent if and only if the corresponding blocks intersect in 3 points, is isomorphic to G5 . 2.1.3. U3 (5) Let G3 be a group isomorphic to the unitary group U3 (5). The group G3 possesses 8 maximal subgroups, up to conjugation (Table 7). Table 7. Maximal subgroups of the group U3 (5) No

Max. sub.

Order

M3,1

A7

2520

50

M3,2 M3,3 M3,4

A7 A7 : Z5 ) : Z8

2520 2520 1000

50 50 126

(E25

Index

M3,5

A6 .Z2

720

175

M3,6 M3,7

A6 .Z2

720

175

A6 .Z2

720

175

M3,8

Z2 .A5 .Z2

240

525

We constructed 3 block designs and 3 strongly regular graphs from the group G3 (Table 8 and Table 9).

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Table 8. Block designs constructed from the group U3 (5) No

Structure

Parameters

Aut. group

D10

D(G3 , M3,1 , M3,5 ; A5 , A6 )

2-(50, 14, 13)

U3 (5) : Z2

D11

D(G3 , M3,4 , M3,8 ; Z5 )

2-(126, 6, 1)

U3 (5) : S3

D12

D(G3 , M3,4 , M3,5 ; Z5 : Z4 )

2-(126, 36, 14)

U3 (5) : Z2

Table 9. Strongly regular graphs constructed from the group U3 (5) No

Structure

Parameters

Aut. group

G6

G(G3 , M3,1 ; A6 )

(50, 7, 0, 1)

U3 (5) : Z2

G7 G8

G(G3 , M3,5 ; D10 ) G(G3 , M3,8 ; Z5 )

(175, 72, 20, 36) (525, 144, 48, 36)

U3 (5) : Z2 U3 (5) : S3

The structure S3 = D(G3 , M3,8 , M3,8 ; Z3 : E4 ) is a semi-symmetric design with parameters (525, 20, (1)), and Aut(S3 ) ∼ = U3 (5) : S3 . = AutU3 (5) ∼ Let M1 and M be the incidence matrices of D11 and S1 , respectively, and I126 be the identity matrix of order 126. Then the matrix P =

I126 M1

M1T M

!

is the incidence matrix of the Desarguesian projective plane P G(2, 25), i.e., a symmetric 2-(651, 26, 1) design. Aut(P G(2, 25)) ∼ = P ΓL3 (25), of order 304668000000. D1 is the Hermitian unital in P G(2, 25). Notes on constructed structures: • D11 is a block design with blocks intersection sizes 1 and 0, and its block graph is a strongly regular graph with parameters (525, 144, 48, 36) isomorphic to the graph G8 . • The design D10 is a derived design of the Higman design 2-(176, 50, 14) (see [41]) and is isomorphic to designs D(G3 , M3,1 , M3,6 ; A5 , A6 ) and D(G3 , M3,1 , M3,7 ; A5 , A6 ). • The design D12 is a residual design of the Higman design 2-(176, 50, 14) (see [41]) and is isomorphic to designs D(G3 , M3,4 , M3,6 ; Z5 : Z4 ) and D(G3 , M3,4 , M3,7 ; Z5 : Z4 ). • The graph G6 is the unique strongly regular graph with these parameters, i.e., the Hoffman-Singleton graph (see [8]) and is isomorphic to graphs G(G3 , M3,2 ; A6 ) and G(G3 , M3,3 ; A6 ). G6 is rank-3 graph obtainable from the representation of degree 50 of the group U3 (5). • The graph G7 is the graph whose vertices are edges of the Hoffman-Singleton graph G2 , two vertices being adjacent if their distance is two (see [41]) and is isomorphic to graphs G(G3 , M3,6 ; D10 ) and G(G3 , M3,7 ; D10 ). • The graph G7 can be constructed from the designs D10 and D12 . Any two blocks of D10 intersect in 3, 4, or 8 points. The graph which has as its vertices the blocks of D10 , two vertices being adjacent if and only if the corresponding blocks intersect in three points, is isomorphic to G7 . Denote this graph by G(D10 , {3, 4, 8}; 3). The graph G(D12 , {6, 10, 11}; 11) is also isomorphic to G3 .

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2.1.4. U3 (7) Let G4 be a group isomorphic to the unitary group U3 (7), i.e. simple group of order 5663616. L1 ∼ = (E49 : Z7 ) : Z48 and L2 ∼ = Z2 .(L2 (7) × Z4 ).Z2 are maximal subgroups of the group G4 of index 344 and 2107, respectively. One can check that Lx1 ∩ Ly2 ∼ = Z7 : Z48 or Z8 for all x, y, ∈ G4 , and D13 = D(G4 , L1 , L2 ; Z7 : Z48 ) is a block design with parameters 2-(344, 8, 1) and full automorphism group isomorphic to the group AutG4 ∼ = U3 (7) : Z2 . The intersection of two distinct elements of cclG4 (L2 ) is isomorphic to Z7 , Z8 , or Z8 × Z8 . S4 = D(U3 (7), L2 , L2 ; Z4 × Z4 ) is a semi-symmetric design with parameters (2107, 42, (1)). Let M1 and M be the incidence matrices of the block design D13 and the semisymmetric design S4 , respectively, and I344 be the identity matrix of order 344. Then the matrix ! I344 M1T P = M1 M is the incidence matrix of the Desarguesian projective plane P G(2, 49) having P ΓL3 (49) as the full automorphism group. The matrix P is a symmetric matrix, so the design D13 is the Hermitian unital in P G(2, 49). 2.1.5. Conjecture We use these computations to conjecture that from any simple group of type U3 (q), by defining incidence structures on the conjugacy classes of maximal subgroups, one can construct a Hermitian unital 2-(q 3 + 1, q + 1, 1) and a semi-symmetric design (q 4 − q 3 + q 2 , q 2 − q, (1)) having Aut(U3 (q)) as an automorphism group, and that the unital can be used to construct a Desarguesian projective plane P G(2, q 2 ) (in the way presented in this paper). 2.2. Structures constructed from unitary groups U4 (2) and U4 (3) 2.2.1. U4 (2) Let G5 be a group isomorphic to the unitary group U4 (2). The group G5 possesses five maximal subgroups, up to conjugation (Table 10). Table 10. Maximal subgroups of the group U4 (2) No

Max. sub.

Order

Index

M5,1

E16 : A5

960

27

M5,2

S6 E27 : S4

720 648

36 40

(E9 : Z3 ) : SL2 (3) Z2 .(A4 × A4 ).Z2

648 576

40 45

M5,3 M5,4 M5,5

We give a list of all constructed structures (Table 11 and Table12). Notes on constructed structures: • The graph G9 is a unique strongly regular graph with parameters (27,10,1,5).

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Table 11. Block designs constructed from the group U4 (2) No

Structure

Parameters

Aut. group

D14

D(G5 , M5,2 , M5,2 ; S3 × S3 )

2-(36, 15, 6)

U4 (2) : Z2

D15

D(G5 , M5,3 , M5,3 ; E9 : S3 , M5,3 )

2-(40, 13, 4)

P ΓL4 (3)

D16

D(G5 , M5,4 , M5,4 ; E9 × S3 , M5,4 )

2-(40, 13, 4)

U4 (2) : Z2

D17

D(G5 , M5,5 , M5,5 ; E4 × A4 )

2-(45, 12, 3)

U4 (2) : Z2

Table 12. Strongly regular graphs constructed from the group U (4, 2) No

Structure

Parameters

Aut. group

G9 G10

G(G5 , M5,1 ; #96.42d 1 ) G(G5 , M5,2 ; S3 × S3 )

(27, 10, 1, 5) (36, 15, 6, 6)

U4 (2) : Z2 U4 (2) : Z2

G11

G(G5 , M5,3 ; E9 : S3 )

(40, 12, 2, 4)

U4 (2) : Z2

G12

G(G5 , M5,4 ; E9 × S3 )

(40, 12, 2, 4)

U4 (2) : Z2

G13

G(G5 , M5,5 ; E4 × A4 )

(45, 12, 3, 3)

U4 (2) : Z2

• The incidence matrix of the design D14 is a symmetric matrix with zero diagonal. Therefore, the design D14 is self-dual and its incidence matrix is an adjacency matrix of a SRG(36, 15, 6, 6) isomorphic to the graph G10 . • A computer-free construction of the design D14 can be found in [30]. • The incidence matrix M of the design D15 is symmetric matrix with 1 everywhere on the diagonal. Therefore, the design D15 is self-dual and the matrix M − I is an adjacency matrix of a SRG(40, 12, 2, 4) isomorphic to the graph G11 . • The design D15 is isomorphic to the design described in [50] and [22]. This design can also be obtained from the point graph of a generalized quadrangle (see [28]). • The incidence matrix M of the design D16 is a symmetric matrix with 1 everywhere on the diagonal. Therefore, the design D16 is self-dual and matrix M − I is an adjacency matrix of a SRG(40, 12, 2, 4) isomorphic to the graph G12 . • The design D16 is the point-hyperplane design in the projective geometry P G(3, 3), which can also be obtained from the point graph of a generalized quadrangle. • The incidence matrix of the design D17 is a symmetric matrix with zero diagonal. Therefore, the design D17 is self-dual and its matrix is an adjacency matrix of a SRG(45, 12, 3, 3) isomorphic to the graph G13 . • The design D17 is isomorphic to the one described in [21] and [39]. 2.2.2. U4 (3) Let G6 be a group isomorphic to the unitary group U4 (3). The group G6 possesses 16 maximal subgroups, up to conjugation (Table 13). Strongly regular graphs constructed from the group G6 are listed in the Table 14. Notes on constructed strongly regular graphs: • The graphs G14 , G15 , G16 and G17 are rank-3 graphs constructed from the rank-3 representation of the group U4 (3) of degrees 112, 126, 162 and 280 respectively. 2.3. Structures constructed from linear groups L2 (49) Let G7 be a group isomorphic to the linear group L2 (49). The group G7 possesses 7 maximal subgroups, up to conjugation (Table 15).

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Table 13. Maximal subgroups of the group U4 (3) No

Max. sub.

Order

Index

No

Max. sub.

Order

Index

M6,1 M6,2

E81 : A6 U4 (2)

29160 25920

112 126

M6,9 M6,10

E16 : A6 A7

5760 2520

567 1296

M6,3

U4 (2)

25920

126

M6,11

A7

2520

1296

M6,4

L3 (4)

20160

162

M6,12

A7

2520

1296

M6,5

L3 (4)

20160

162

M6,13

A7

2520

1296

M6,6 M6,7

(Ex+ 243 : Q2 ).S3 U3 (3)

11664 6048

280 540

M6,14 M6,15

((E8 .Z12 ) : Z6 ) : Z2 A6 .Z2

1152 720

2835 4536

M6,8

E16 : A6

5760

567

M6,16

A6 .Z2

720

4536

Table 14. Strongly regular graphs constructed from the group U4 (3) No

Structure

Parameters

Aut. group

G14

G(G6 , M6,1 ; Ex+ 273 : Z4 )

(112, 30, 2, 10)

U4 (3) : D4

G15 G16

G(G6 , M6,2 ; (E8 .Z12 ) : Z6 ) G(G6 , M6,4 ; A6 )

(126, 45, 12, 18) (162, 56, 10, 24)

U4 (3) : Z2 ) : Z2 (U4 (3) : Z2 ) : Z2

G17

G(G6 , M6,6 ; E81 : Z4 )

(280, 36, 8, 4)

U4 (3) : D4

G18

G(G6 , M6,7 ; E9 : Z3 )

(540, 224, 88, 96)

U4 (3) : D4

Table 15. Maximal subgroups of the group L( 49) No

Max. sub.

Order

M7,1 M7,2

E49 : Z24

1176

Index 50

P GL2 (7)

336

175

M7,3 M7,4

P GL2 (7) A5

336 60

175 980

M7,5

A5

60

980

M7,6 M7,7

D50 D48

50 48

1176 1225

From the group G7 we constructed 2 block designs (Table 16). Table 16. Block designs constructed from the group L2 (49) No

Structure

Parameters

Aut. group

D18

D(G7 , M7,1 , M7,2 ; Z7 : Z6 )

2-(50, 8, 4)

L2 (49) : Z2

D19

D(G7 , M7,1 , M7,4 ; Z3 )

2-(50, 20, 152)

L2 (49) : Z2

The design D18 is isomorphic to the design D(G7 , M7,1 , M7,3 ; Z7 : Z6 ) and the design D19 is isomorphic to the design D(G7 , M7,1 , M7,5 ; Z3 ). 3. Codes from the designs and graphs It is well-known that combinatorial design theory and coding theory are closely related. Certain combinatorial structures have been used to construct good codes. Such example structures include balanced incomplete block designs, symmetric designs, resolvable designs, strongly regular graphs, etc.

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3.1. Some codes from the designs and graphs of the unitary groups U3 (3), U3 (5), U4 (2) and U4 (3) For the binary codes associated with the unitary group U3 (3) the reader should consult the work of Broke in [6], and for some codes invariant under the unitary group U3 (5), see [41,56]. Binary codes invariant under U4 (2) have been studied in [7]. Work determining all binary codes invariant under the groups U3 (4) and U3 (5) is currently in progress. As an illustration of the links with codes, in this section, we examine some linear codes associated with the combinatorial structures described in Sections 2.2.1, 2.2.2 and 2.3 respectively, and which are obtained from the row span over the finite field of the corresponding incidence or adjacency matrices. In particular we examine results on some binary, and ternary codes invariant under the groups U4 (2), U4 (3) and L2 (49) respectively, as given for example in [45,16] and in [46]. Some interesting self-dual and self-orthogonal codes are obtained from these structures. A code with the property that its weight enumerator coincides with its MacWilliams transform is called formally self-dual. A linear code is termed isodual if it is equivalent to its dual. Thus an isodual code is automatically formally self-dual. A self-dual code is isodual and even, however a formally self-dual code need not be isodual. One motivation for this study is that codes associated with strongly regular graphs admit an efficient decoding method, known as majority decoding [55]. Moreover, strongly regular graphs have been used recently with success in the construction of self-dual codes (see [27]). In this section we give an example of this association by constructing self-dual [72, 36, 8], [80, 40, 12], [80, 40, 8] codes from some designs and isodual [54, 27, 8]2 , [90, 45, 12]2 , [224, 112, 6]2 codes from some strongly regular graphs. Furthermore, we use the properties of the designs and the graphs and their geometry to gain some insight into the nature of possible codewords, particularly those of minimum weight. 3.1.1. Codes of graphs (designs) from U4 (2) Notice that the simple group U4 (2) acts as a rank-3 primitive group in all its representations [12], thus producing rank-3 graphs. The reader should be aware that rank-3 graphs give rise to strongly regular graphs. However strongly regular graphs do not give rise to rank-3 graphs. In this section we examine the properties of the linear codes constructed from the strongly regular graphs Gi where 9 ≤ i ≤ 13 as presented in [Section 2.2.1,Table 12]. Notice that with exception of G9 all other graphs in Table 12 possess the property that they or their complements (the complements of the graphs G11 and G12 are graphs with parameters (40, 27, 18, 18)) are such that λ = µ, so we may associate with every (n, k, λ, µ) graph a 2-(n, k, λ) design and thus construct the codes spanned by the incidence matrices of such designs. Since these designs (or their complements) are self-orthogonal, the codes spanned by the block-point incidence matrices are self-orthogonal [53]. We adopted this view in [45] and surveyed the interplay between this very special class of designs and self-orthogonal codes. Since the orders of these designs is divisible by 3, we only examined ternary codes obtained from the row span of the incidence matrices of the designs D14 , D15 , D16 and D17 or those of their respective complements, as given in [Section 2.2.1,Table 11]. We establish some properties of these codes and the nature of some classes of codewords. Some of the codes are optimal or near optimal for the given length and dimension. The dual codes of some designs and

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those of some complementary designs admit majority logic decoding. Notice that the design D14 belongs to a series with parameters v = 4m2 , k = m(2m − 1), λ = m(m − 1), n = m2 (see [4], p. 622 with m = 3). In [45] we proved the following results: Proposition 1 (i) D14 is a self-orthogonal design. (ii) CD14 is a [36, 15, 9]3 self-orthogonal code. (iii) CD14 ⊥ is a [36, 21, 6]3 with 240 words of weight 6. Moreover, 1 ∈ CD14 and in CD14 ⊥ . (iv) Aut(D14 ) = Aut(CD14 ) ∼ = U4 (2):Z2 . Proof: Since |Bi ∩ Bj | ≡ |Bk | ≡ 0(mod 3), (where i, j, k ∈ {1, . . . , b}, i = j and b and k are respectively the number of blocks and the block size), we deduce that D14 is a self-orthogonal design. Hence the block-point incidence matrix of D36 spans a selforthogonal code CD14 of length 36 [53]. Since the block size of D14 is divisible by 3 we have that 1 ∈ CD14 ⊥ . Now, from [30, Theorem 1] we have that Aut(D14 ) ∼ = P ΓU4 (2). Since Aut(D14 ) ⊆ Aut(CD14 ) and |Aut(CD14 )| = |P ΓU4 (2)|, the result follows. In particular CD14 contains the vector 1. The minimum distance 9 can be deduced from the weight enumerator for this code which is as follows: WCD14 = 1 + 80 x9 + 3240 x12 + 43632 x15 + 693600 x18 + 3355344 x21 + 5992110 x24 + 3654320 x27 + 587736 x30 + 18360 x33 + 484 x36 . Computation with Magma [3] show that dim(CD14 ) = 15 and that CD14 ⊥ has minimum weight 6.  Since 1 ∈ CD14 it follows that the code of the complementary design 2-(36, 21, 12) is CD14 . Following the above, here we look at the ternary codes of the complementary de˜ 15 and D ˜ 16 respectively. Observe that these designs are signs of D15 and D16 namely D ˜ non-isomorphic, and that D15 , i.e., a 2-(40, 27, 18) design, is the design of points and lines of the projective geometry P G(2, 3). These designs have the group L4 (3):(Z2 )1 , as their full automorphism group. Notice that the group L4 (3) has three involutory outer automorphisms (see [12]), namely (Z2 )1 , (Z2 )2 and (Z2 )3 . The groups L4 (3):(Z2 )1 , L4 (3):(Z2 )2 and L4 (3):(Z2 )3 are non-isomorphic. Notice that Aut(U4 (2)) = U4 (2):Z2 ≤ Aut(D15 ) ∼ = L4 (3):(Z2 )1 . From the Atlas [12] we have that U4 (2):Z2 is a maximal subgroup of L4 (3):21 of index 234. Theorem 3 The linear group L4 (3):(Z2 )1 is the automorphism group of the [40, 10]3 ˜ 15 . D ˜ 15 is a self-orthogonal design. The code C ˜ is ternary code CD˜ 15 derived from D D15 self-orthogonal, with minimum distance 18. Its dual is a [40, 30, 4]3 with 260 words of weight 4. Moreover 1 ∈ CD˜ 15 ⊥ . ˜ 15 follows using an argument similar to that in the proof Proof: Self orthogonality of D of Proposition 1, hence the self-orthogonality of CD˜ 15 . The minimum distance 18 can be deduced from the weight enumerator for this code which is as follows:

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WCD˜

15

= 1 + 1560 x18 + 21060 x24 + 18800 x27 + 16848 x30 + 780 x36 .

Computation with Magma shows that dim(CD˜ 15 ) = 10 and that CD˜ 15 ⊥ has minimum weight 4.  Remark 3 The code and groups found above can be described geometrically: with the notation of the propositions, the [40, 30, 4]3 code CD˜ 15 ⊥ is in fact the code of the 2(40, 4, 1) design of points and lines in the projective geometry P G(2, 3); the automorphism group of the design is P ΓL3 (3) ∼ = L4 (3):(Z2 )1 , by the fundamental theorem of projective geometry. The 260 words of weight 4 are the incidence vectors of the lines, and their scalar multiples. The code CD˜ 15 ⊥ is in fact a projective generalized Reed-Muller code (see [1, Chapter 5]). The words of weight 18 in CD˜ 15 can also be described geometrically, i.e., they are the differences of the incidence vectors of two planes of order 3 in P G(2, 3). These planes meet in a line, i.e., four points, so the weight of the difference of incidence vectors is 18. The code CD˜ 15 is an optimal code, and its dual code CD˜ 15 ⊥ has minimum distance ˜ 15 , only 1 less than the optimal. The code of the complementary design 2-(40, 13, 4) of D which is obtained from CD˜ 15 by adding the all-one vector, is a [40, 11, 13]3 . This code is far from being optimal. However the dual of this code is an optimal [40, 29, 6]3 code. The rows of the incidence matrix of the design D15 can be used as orthogonal parity checks that allow majority decoding of the code [40, 29, 6]3 up to its full error-correcting capacity. The following proposition can now be proved Proposition 2 The code [40, 29, 6]3 can correct up to 2 errors by majority decoding. Applying the 5Rudolph’s decoding algorithm [47] for the design D15 we have that 4Proof: 5 4 13+4−1 r+λ−1 = = 2, and so the result.  2λ 2·4 ˜ 16 we have the following Associated with D ˜ 16 is a self-orthogonal design. Proposition 3 (i) D (ii) CD˜ 16 is a [40, 14, 12]3 self-orthogonal code. (iii) 1 ∈ CD˜ 16 ⊥ . (iv) CD˜ 16 ⊥ is a [40, 26, 4]3 with 80 words of weight 4. ˜ 16 ) = Aut(C ˜ ) ∼ (v) Aut(D D16 = U4 (2):Z2 . The weight enumerator for CD˜ 16 is as follows: WCD˜

16

= 1 + 540 x12 + 3600 x15 + 39360 x18 + 305280 x21 + 1228320 x24 + 1982240 x27 + 1017648 x30 + 193680 x33 + 11580 x36 + 720 x39 .

˜ 16 = Aut D ˜ 15 we have that D ˜ 16  D ˜ 15 . The code C ˜ has Remark 4 Since Aut D D16 minimum distance only 3 less than the optimal. The same occurs for its dual code CD˜ 16 ⊥ . The code CD16 , of D16 which is obtained from CD˜ 16 by adding the all-one vector, is a [40, 15, 10]3 . This code is far from being optimal. The dual of this code a [40, 25, 6]3

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has minimum distance only 2 less than the optimal. However [40, 25, 6]3 can be used to correct up to 2 errors by majority decoding. We now look at the ternary codes of D17 . Notice that symmetric 2-(45, 12, 3) designs belong to the series with parameters v = q l+1



1+

q l+1 − 1 q−1



, k = ql

ql − 1 q l+1 − 1 and λ = q l , q−1 q−1

where q is any prime power and l is any positive integer (see [4], p. 622 with q = 3 and l = 1). Proposition 4 (i) D17 is a self-orthogonal design. (ii) CD17 is a [45, 15, 12]3 self-orthogonal code. (iii) 1 ∈ CD17 and 1 ∈ CD17 ⊥ . (iv) CD17 ⊥ is a [45, 30, 6]3 with 1200 words of weight 6. (v) Aut(D17 ) = Aut(CD17 ) ∼ = U4 (2):Z2 . The weight enumerator of CD17 is given by WCD17 = 1 + 90 x12 + 1152 x15 + 8660 x18 + 92340 x21 + 952020 x24 + 3394640 x27 + 5270400 x30 + 3712770 x33 + 850170 x36 + 63360 x39 + 3060 x42 + 244 x45 . The words of minimum weight in CD17 are the incidence vectors of the blocks of the design and their scalar multiples. The code CD17 is not optimal. However its dual code CD17 ⊥ has minimum distance only 1 less than the optimal. Since 1 ∈ CD17 it follows that the code of the complementary design 2-(45, 33, 24) is CD17 . The dual code CD17 ⊥ is a [45, 30, 6]3 , and the rows of the incidence matrix of D17 can be used as orthogonal parity checks that allow majority decoding of CD17 ⊥ up to its full error-correcting capacity. 3.1.2. Self-dual [72, 36, 8]2 , [80, 40, 12]2 and [80, 40, 8]2 codes The existence of self-dual [72, 36, 16] code is an important coding theory question. It is shown in [26] that a code with these parameters could be found from Hadamard matrices of order 36 with a trivial group or with automorphisms of order 2, 3, 5 or 7. This is one motivation for our study of the codes given below. Using a well-known construction, such as for example that given in [5] we construct a self-dual type II [72, 36, 8]2 code associated with the design D14 . Theorem 4 ([5, Theorem 2]) Let A be the incidence matrix of a symmetric 2-(v, k, λ) design with k − λ odd. Then: (i) if k ≡ 3(mod 4), then the code with generator matrix (I A) is a doubly-even self-dual [2v, v] code. (ii) if k ≡ 2(mod 4), then the code with generator matrix

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⎛

⎞ 1 ... 1 0 ⎜ 1 ⎟ ⎟ ⎜ ⎝ Iv+1 A . . . ⎠ 1 is a doubly-even self-dual [2v + 2, v + 1] code. Using Theorem  4 a generator matrix of a double-even self-dual code of length 72 can be obtained as A I36 , so we construct a type II [72, 36, 8]2 self-dual code denoted PT .

  Proposition 5 The binary code PT of A I36 is a formally self-dual type II [72, 36, 8]2 code, with automorphism group isomorphic to E32768 :S6 (2). The weight enumerator of PT is as follows: WPT (x) = 1 + 945 x8 + 30576 x12 + 535932 x16 + 17267040 x20 + 455965020 x24 + 4438423440 x28 + 16506508662 x32 + 25882013504 x36 + 16506508662 x40 + 4438423440 x44 + 455965020 x48 + 17267040 x52 + 535932 x56 + 30576 x60 + 945 x64 + x72 .

Similarly, using Theorem 4 we constructed self-dual type II [80, 40, 12]2 and [80, 40, 8]2 codes from the designs D15 and D16 respectively. So we have Proposition 6 The binary code of D15 is a formally self-dual type II [80, 40, 12]2 code, with 4160 words of weight 12, and automorphism group isomorphic to L4 (3):(E4 ). Moreover the binary code of D16 is a formally self-dual type II [80, 40, 8]2 code, with 270 words of weight 8, and automorphism group isomorphic to Z2 .U4 (2):Z2 . Remark 5 The binary code of D17 is an isodual [90, 45, 12] with 1160 codewords of weight 12, and automorphism group isomorphic to U4 (2):Z2 . 3.1.3. Codes from U4 (3) Similar to the analysis provided in Section 3.1.1, here we look at the codes obtained from strongly regular graphs constructed from the simple unitary group U4 (3). These graphs are in fact rank-3 graphs. A study of the binary codes of strongly regular graphs, including some known graphs on fewer than 45 vertices has been undertaken in [29]. As discussed in [Section 2.2.2,Table 13] using Theorem 2, from the conjugacy classes of maximal subgroups of the simple unitary group U4 (3) we obtain strongly regular graphs Gi , where 14 ≤ i ≤ 17. A code CGi of a graph Gi is the code of its (0, 1)-adjacency matrix. The dimension of CGi is equal to the p-rank of its adjacency matrix, i.e., the rank of Gi regarded as a matrix over GF (p). This section discusses the codes of these graphs as presented in [16] without an association to 2-designs. Notice first that the unitary group U4 (3) is a maximal subgroup of the sporadic simple group Mc L discovered

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by J. McLaughlin [32]. It was shown by McLaughlin that there exists a regular graph G = (Ω, E) with 275 vertices possessing a transitive automorphism group Aut(G) ∼ = Mc L:2, with Mc L a simple group of order 898128000. The McLaughlin graph G is a rank-3 graph of valency 112 on 275 points. The stabilizer of a point in Mc L is a maximal subgroup isomorphic to U4 (3). The orbits under this action have lengths 1, 112 and 162, respectively. The McLaughlin graph contains many induced subgraphs which are again strongly regular. The graphs denoted G14 and G16 in Section 2.2.2 which are the first and second subconstituents of the McLaughlin graph are one such example. The uniqueness of these graphs was proved in [9,11]. We now look at the codes of G14 and G16 respectively. Proposition 7 (i) CG14 is a [112, 22, 30]2 self-orthogonal code. (ii) 1 ∈ CG14 . (iii) CG⊥14 is a self-complementary [112, 90, 6]2 code with 5040 words of weight 6. (iv) Aut(G14 ) = Aut(CG14 ) ∼ = U4 (3)·D8 . (v) CG16 is a [162, 20, 56]2 self-orthogonal doubly-even code. (vi) CG⊥16 is a self-complementary [162, 142, 6]2 code with 86562 words of weight 6. Moreover 1 ∈ CG⊥16 . (vii) Aut(G16 ) = Aut(CG16 ) ∼ = U4 (3) · (22 )133 . (viii) U4 (3) acts irreducibly on CG16 as a GF (2) module. Uniqueness of the graphs G15 and G17 is not known. The codes of these graphs are given in Proposition 8 (i) CG15 is a [126, 21, 36]3 self-orthogonal code. (ii) CG⊥15 is a [126, 105, 6]3 code with 23250 words of weight 6. (iii) 1 ∈ CG15 . (iv) Aut(G15 ) = Aut(CG15 ) ∼ = U4 (3) · (22 )122 . (v) CG17 is a [280, 70, 36]2 self-orthogonal with 280 words of weight 36. (vi) CG⊥17 is a [280, 210, 8]2 . (vii) 1 ∈ CG17 . (viii) Aut(G17 ) = Aut(CG17 ) ∼ = U4 (3)·D8 . Remark 6 The codes of the complements of the graphs Gi , where 14 ≤ i ≤ 17, have been examined in [16]. We have constructed an isodual [224, 112, 6]2 code from the adjacency matrix of the graph G14 . 3.2. Codes from L2 (49) In closing we consider the results presented in [Section 2.3,Table 15], and look at the codes obtained from the designs D18 and D19 . In [46], a discussion of the properties of the codes of these designs is given, and links with isodual codes are established. Work related with isodual codes could be found in [2,33,31,52]. Isodual codes have many practical applications and their mathematical structure provides useful information for computing their support weight enumerators. Proposition 9 The linear group L2 (49):Z2 is the automorphism group of the [50, 25, 8]2 code CD18 derived from the 2-(50, 8, 4) design D18 . The code CD18 is also the code of the design D19 . Moreover, CD18 is an isodual code.

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The weight distribution of CD18 is given below (Table 17). Table 17. Weight distribution of CD18 i

Ai

0, 50

1

8, 42

175

12, 38

9800

16, 34

287875

18, 32

1102500

20, 30

2808190

22, 28

5225500

24, 26

7292425

The rows of the incidence matrix of the 2-(50, 8, 4) design can be used as orthogonal parity checks that allow majority decoding of the code [50, 25, 8]2 up to its full errorcorrecting capacity.

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-253

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Matrices for graphs, designs and codes Willem H. HAEMERS Department of Econometrics and OR, Tilburg University, The Netherlands Abstract. The adjacency matrix of a graph can be interpreted as the incidence matrix of a design, or as the generator matrix of a binary code. Here these relations play a central role. We consider graphs for which the corresponding design is a (symmetric) block design or (group) divisible design. Such graphs are strongly regular (in case of a block design) or very similar to a strongly regular graph (in case of a divisible design). Many constructions and properties for these kind of graphs are obtained. We also consider the binary code of a strongly regular graph, work out some theory and give several examples. Keywords. Block designs, divisible designs, strongly regular graphs, Seidel switching, Hadamard matrices, binary codes.

1. Introduction Central is this reader is the interplay between graphs and designs. We start with a preliminary chapter on strongly regular graphs, block designs and their interplay. Then we will look at binary codes generated by the adjacency matrix of a strongly regular graph. This section is mainly based on [24]. The third part, based on [21], is devoted to a more recent development on graphs that are related to divisible designs. We introduce basic concepts as block designs, strongly regular graphs and Hadamard matrices, but we assume basic knowledge of algebra and graph theory. Some useful general references are [4,16,29,39].

2. Graphs and designs 2.1. Designs A block design with parameters (v, k, λ) is a finite point set P of cardinality v, and a collection B of subsets (called blocks) of P, such that: (i) Each block has cardinality k (2 ≤ k ≤ v − 1). (ii) Each (unordered) pair of points occurs in exactly λ blocks. A block design with parameters (v, k, λ) is also called a 2-(v, k, λ) design. The incidence matrix N of such a design is the (0, 1) matrix with rows indexed by the points, and columns indexed by the blocks, such that Nij = 1 if point i is in block j, and Nij = 0 otherwise. The following result is a straightforward translation of the definition into matrix language. (as usual, J stands for an all-ones matrix, and 1 for an all-ones vector).

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Proposition 2.1 A (0, 1) matrix N is the incidence matrix of a 2-(v, k, λ) design if and only if N ⊤1 = k1 and N N ⊤ = λJ + D , for some diagonal matrix D. Theorem 2.1 Suppose (P, B) is a 2-(v, k, λ) design with incidence matrix N , then (i) each point is incident with r = λ(v − 1)/(k − 1) blocks, that is N 1 = r1, and N N ⊤ = λJ + (r − λ)I, (ii) the number of blocks equals b = vr/k, that is N has b columns, (iii) b ≥ v with equality if and only if N ⊤ is the incidence matrix of a 2-(v, k, λ) design. Proof. (i): Fix a point z ∈ P. By use of ii of the above definition, the number of pairs (x, a) with x ∈ P, x = z and a ∈ B, z ∈ a equals λ(v − 1). On the other hand it is equal to k − 1 times the number of blocks containing z. Equation (ii) follows by counting the number of pairs (x, a) with x ∈ P, a ∈ B, a ∈ a (that is, the number of ones in N ). (iii): From (i) and Proposition 2.1 it follows that R=

λ 1 N⊤+ J r−λ r(r − λ)

is a right inverse of N . Therefore N has rank v, and hence b ≥ v. Moreover, if b = v, then r = k, R = N −1 and we have I = N −1 N , which leads to N ⊤N = (k − λ)I + λJ. By (i) we have N 1 = k1, hence N ⊤ is the incidence matrix of a 2-(v, k, λ) design by Proposition 2.1. ⊔ ⊓ A 2-(v, k, 1) design is also called a Steiner 2-design. A block design with b = v is called symmetric. The dual of a design with incidence matrix N is the structure with incidence matrix N ⊤. Theorem 2.1(ii) states that the dual of a symmetric design is again a symmetric design with the same parameters. In terms of the original design, it means that any two distinct blocks intersect in the same number of points. In general, the size of the intersection of two distinct blocks can vary. If in a block design these numbers take only two values, we call the design quasi-symmetric. Obviously, two blocks in a Steiner 2-design cannot have more than one points in common, so it is symmetric, or quasi-symmetric. Note that if N is the incidence matrix of a 2-(v, k, λ) design (P, B), then J − N represents a 2-(v, v − k, b − 2v + λ) design, called the complement of (P, B). Moreover, if N is symmetric (or quasi-symmetric), the so is the complement. Many examples of block designs come from geometries over a finite field Fq . For example the points end the lines in in projective space of dimension n over Fq give a 2-(q n + q n−1 + . . . + q + 1, q + 1, 1) design. Because λ = 1, it is Steiner 2-design, and therefore quasi-symmetric, or symmetric. The design is a symmetric if and only if n = 2. Such a design is called a projective plane of order q. The smallest case q = 2 gives the famous Fano plane. Another family of examples comes from Hadamard matrices. An m × m matrix H is a Hadamard matrix (of order m) if every entry is 1 or −1, and HH ⊤ = mI. In other 1 words, H −1 = m H ⊤ hence H ⊤H = mI. If a row or a column of a Hadamard matrix is

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multiplied by −1, the matrix remains a Hadamard matrix. Therefore we can accomplish that the first row and column consist of ones only. If we then delete the first row and column we obtain a (m − 1) × (m − 1) matrix C, often called the core of H (with respect to the first row and column). It follows straightforwardly that a core C of a Hadamard matrix satisfies CC ⊤ = C ⊤C = mI − J, and C1 = C ⊤1 = −1. This implies that N = 21 (C + J) satisfies N ⊤1 = ( 12 m − 1)1 and N N ⊤ = 14 mI + ( 41 m − 1)J, that is, N is the incidence matrix of a 2-(m − 1, 12 m − 1, 14 m − 1) design (provided m > 2). Note that this implies that if m > 2, then m is divisible by 4. A Hadamard matrix H is regular if H has constant √ row and column sum (ℓ say). From HH ⊤ = mI we get that ℓ2 = m, so ℓ = ± m, and m is a square. If H is a regular Hadamard matrix, the we easily have that N = 21 (H + J) is the incidence matrix of a symmetric 2-(m, (m + ℓ)/2, (m + 2ℓ)/4) design. Examples of Hadamard matrices are: ⎤ ⎡ −1 −1 −1 1 ! ⎢ −1 −1 1 −1 ⎥ 1 1 ⎥ and ⎢ ⎣ −1 1 −1 −1 ⎦ . 1 −1 1 −1 −1 −1 One easily verifies that, if H1 and H2 are Hadamard matrices, then so is the Kronecker product H1 ⊗H2 . Moreover, if H1 and H2 are regular, then so is H1 ⊗H2 . With the above examples (note that the second one is regular) we can construct Hadamard matrices of order m = 2i , and regular ones of order 4i for i ≥ 0. Many more constructions for Hadamard matrices and block designs are known. Some general references are [6] and [16], Chapter V. 2.2. Strongly regular graphs A strongly regular graph with parameters (v, k, λ, µ) (often denoted by SRG(v, k, λ, µ)) is a (simple undirected and loopless) graph of order v satisfying: (i) each vertex is adjacent to k (1 ≤ k ≤ v − 2) vertices, (ii) for each pair of adjacent vertices there are λ vertices adjacent to both, (iii) for each pair of non-adjacent vertices there are µ vertices adjacent to both. For example, the pentagon is strongly regular with parameters (v, k, λ, µ) = (5, 2, 0, 1). One easily verifies that a graph Γ is strongly regular with parameters (v, k, λ, µ) if and only if its complement Γ is strongly regular with parameters (v, v − k − 1, v − 2k + µ − 2, v − 2k + λ). The line graph of the complete graph of order m, known as the triangular graph T (m), is strongly regular with parameters ( 12 m(m − 1), 2(m − 2), m − 2, 4). The complement of T (5) has parameters (10, 3, 0, 1). This is the Petersen graph (see Figure 1). A graph Γ satisfying condition (i) is called k-regular. The adjacency matrix of a graph Γ is the symmetric (0, 1) matrix A indexed by the vertices of Γ, where Aij = 1 if i is adjacent to j, and Aij = 0 otherwise. It is well-known and easily seen that A1 = k1 for a k-regular graph, in other words, the adjacency matrix of a k-regular graph has an eigenvalue k with eigenvector 1. Moreover, every other eigenvalue ρ satisfies |ρ| ≤ k, and if Γ is connected, the multiplicity of k equals 1 (see Biggs [7]). For convenience we call an eigenvalue restricted if it has an eigenvector perpendicular to 1. So for a k-regular connected graph the restricted eigenvalues are the eigenvalues different from k.

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✉ ✚❩ ❩ ✚ ✉ ❩ ✚ ✉ ❩✉ ✚ PP✉ ✂❇ ✉✏ ✏ ❇ ❩❩✂ ❇✚✚ ✂✂ ❇ ✂❩ ✚❇ ❇ ✂ ❩❇✉ ✂ ✚ ✉ ❇ ✜ ❭❭✂✂ ❇✜ ✉ ✉ Figure 1. The Petersen graph

Theorem 2.2 For a simple graph Γ of order v, not complete or empty, with adjacency matrix A, the following are equivalent: (i) G is strongly regular with parameters (v, k, λ, µ) for certain integers k, λ, µ, (ii) A2 = (λ − µ)A + (k − µ)I + µJ for certain reals k, λ, µ, (iii) A has precisely two distinct restricted eigenvalues. Proof. The equation in (ii) can be rewritten as A2 = kI + λA + µ(J − I − A). Now (i) ⇔ (ii) is obvious. (ii) ⇒ (iii): Let ρ be a restricted eigenvalue, and u a corresponding eigenvector perpendicular to 1. Then Ju = 0. Multiplying the equation in (ii) on the right by u yields ρ2 = (λ−µ)ρ+(k −µ). This quadratic equation in ρ has two distinct solutions. (Indeed, (λ − µ)2 = 4(µ − k) is impossible since µ ≤ k and λ ≤ k − 1.) (iii) ⇒ (ii): Let r and s be the restricted eigenvalues. Then (A − rI)(A − sI) = αJ for some real number α. So A2 is a linear combination of A, I and J. ⊔ ⊓ As an application, we show that quasi-symmetric block designs give rise to strongly regular graphs. Recall that a quasi-symmetric design is a 2-(v, k, λ) design in which any two distinct blocks meet in either x or y points, for certain fixed x, y. Given this situation, we may define a graph Γ on the set of blocks, and call two blocks adjacent when they meet in x points. Then there exist coefficients α1 , . . . , α7 such that N N ⊤ = α1 I + α2 J, N J = α3 J, JN = α4 J, A = α5 N ⊤N + α6 I + α7 J, where A is the adjacency matrix of the graph Γ. (The αi can be readily expressed in terms of v, k, λ, x, y.) Then Γ is strongly regular by (ii) of the previous theorem. Indeed, from the equations just given it follows straightforwardly that A2 can be expressed as a linear combination of A, I and J. We know that all 2-(v, k, 1) designs are quasi-symmetric. This leads to a substantial family of strongly regular graphs, including the triangular graphs T (m) (derived from the trivial design consisting of all pairs out of an m-set). Theorem 2.3 Let Γ be a strongly regular graph with adjacency matrix A and parameters (v, k, λ, µ). Let r and s (r > s) be the restricted eigenvalues of A and let f and g be their respective multiplicities. Then (i) k(k − 1 − λ) = µ(v − k − 1), (ii) rs = µ − k, r + s = λ − µ, (iii) f, g = 21 (v − 1 ∓ (r+s)(v−1)+2k ). r−s

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(iv) r and s are integers, except perhaps when f = g, (v, k, λ, µ) = (4t + 1, 2t, t − 1, t) for some integer t. Proof. (i) Fix a vertex x of Γ. Let Γ(x) and ∆(x) be the sets of vertices adjacent and nonadjacent to x, respectively. Counting in two ways the number of edges between Γ(x) and ∆(x) yields (i). The equations (ii) are direct consequences of Theorem 2.2(ii), as we saw in the proof. Formula (iii) follows from f + g = v − 1 and 0 = trace A = k + f r + gs = k + 12 (r + s)(f + g) + 21 (r − s)(f − g). Finally, when f = g then one can solve for r and s in (iii) (using (ii)) and find that r and s are rational, and hence integral. But f = g implies (µ − λ)(v − 1) = 2k, which is possible only for µ − λ = 1, v = 2k + 1. ⊔ ⊓ These relations imply restrictions for the possible values of the parameters. Clearly, the right hand sides of (iii) must be positive integers. These are the so-called rationality conditions. As an example of the application of the rationality conditions we can derive the following result due to Hoffman & Singleton [27] Theorem 2.4 Suppose (v, k, 0, 1) is the parameter set of a strongly regular graph. Then (v, k) = (5, 2), (10, 3), (50, 7) or (3250, 57). Proof. The rationality conditions imply that either f = g, which leads to (v, k) = (5, 2), or r − s is an integer dividing (r + s)(v − 1) + 2k. By use of Theorem 1(i)-(ii) we have s = −r − 1, k = r2 + r + 1, v = r4 + 2r3 + 3r2 + 2r + 2, and thus we obtain r = 1, 2 or 7.

⊔ ⊓

The first three possibilities are uniquely realized by the pentagon, the Petersen graph and the Hoffman-Singleton graph. For the last case existence is unknown Except for the rationality conditions, a few other restrictions on the parameters are known. We mention two of them. The Krein conditions [35], can be stated as follows: (r + 1)(k + r + 2rs) ≤ (k + r)(s + 1)2 , (s + 1)(k + s + 2rs) ≤ (k + s)(r + 1)2 . The absolute bound (see Delsarte, Goethals & Seidel [17]) reads, v ≤ f (f + 3)/2, v ≤ g(g + 3)/2. The Krein conditions and the absolute bound are special cases of general inequalities for association schemes, see for example [10]. For constructions and more results on strongly regular graphs we refer to [11], [12], [15], [16], [28], or [36]. 2.3. Neighborhood designs Any graph Γ can be interpreted as a design, by taking the vertices of Γ as points, and the neighborhoods of the vertices as blocks. In other words, the adjacency matrix of Γ is interpreted as the incidence matrix of a design. Let us call such a design the neighborhood design of Γ.

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Consider a strongly regular graph Γ with parameters (v, k, λ, µ). If λ = µ, then any two distinct vertices have exactly λ common neighbors, and the adjacency matrix A of Γ satisfies AA⊤ = A2 = (k − λ)I + λJ . This implies that the neighborhood design of Γ is a symmetric 2-(v, k, λ) design (sometimes called: (v, k, λ) design). Rudvalis [34] has called such a graph a (v, k, λ) graph. If a symmetric design admits a symmetric incidence matrix, the corresponding bijection between points and blocks is called a polarity of the design. The points (and blocks) that correspont to a 1 on the diagonal are the absolute points (blocks) of the polarity. Thus a (v, k, λ) design with a polarity with no absolute points can be interpreted as a (v, k, λ) graph. Similarly, if A is the adjacency matrix of a strongly regular graph with parameters (v, k, λ, λ + 2), then A + I is the incidence matrix of a square 2-(v, k, λ) design, and in this way one obtains precisely the 2-(v, k, λ) designs possessing a polarity with all points absolute. This interplay between graphs and designs turned out to be fruitful for both parts. For example, an easy construction of a symmetric 2-(16, 6, 2) design goes via the 4 × 4 grid, (that is, the line graph of the complete bipartite graph K4,4 , also known as the Lattice graph L(4)), which is a (16, 6, 2) graph. It may happen, however, that two nonisomorphic (v, k, λ) graphs, Γ1 and Γ2 with adjacency matrices A1 and A2 say, give isomorphic designs. Also A1 and A2 +I can represent isomorphic designs. The standard example is given by the two SRG(16, 6, 2, 2)’s (the lattice graph L(4) and the Shrikhande graph) and the unique SRG(16, 5, 0, 2) (the Clebsch graph). The three graphs produce the same symmetric 2-(16, 6, 2) design. Proposition 2.2 If two non-isomorphic (v, k, λ) graphs Γ1 and Γ2 give rise to isomorphic (v, k, λ) designs, then both Γ1 and Γ2 have an involution (that is, an automorphism of order 2). Proof. Let Ai be the adjacency matrix of Γi (i = 1, 2), and assume that the corresponding designs are isomorphic. Then there exist permutation matrices P and Q such that P A1 Q = A2 . Without loss of generality we assume Q = I (otherwise replace A2 by Q⊤A2 Q). The symmetry of A2 gives P A1 = A1 P ⊤, and hence P m A1 = A1 (P m )⊤. If P has even order 2m, then P 2m = I and P m = (P m )⊤ = I. This implies A1 = P 2m A1 = P m A1 (P m )⊤, so P m is an involution. If P has odd order 2m − 1, then A2 = P A1 = P 2m A1 = P m A1 (P m )⊤, so Γ1 and Γ2 are isomorphic graphs. ⊔ ⊓ So, if for example a (v, k, λ) graph Γ has a trivial automorphism group, then any other (v, k, λ) graph not isomorphic to Γ gives a non-isomorphic design. For instance, there exist 16428 (36, 21, 12) graphs. From these graphs, 15127 have a trivial automorphism group (see [37], [30]). So at least 15128 are also non-isomorphic as designs. A large family of (v, k, λ) graphs comes from regular graphical Hadamard matrices. A Hadamard matrix H is graphical if it is symmetric with constant diagonal. Without loss of generality we assume that the diagonal elements are −1 (otherwise we √ replace H by −H). If, in addition, H is regular of order m with row sum ℓ = ± m, then A = 21 (H + J) is the adjacency matrix of an (m, (m + ℓ)/2, (m + 2ℓ)/4) graph. The two smallest regular graphical Hadamard are:

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⎡

⎤

⎡

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⎤

−1 −1 −1 1 −1 1 1 1 ⎢ −1 −1 1 −1 ⎥ ⎢ 1 −1 1 1 ⎥ ⎥ ⎥ ⎢ ⎢ ⎣ 1 1 −1 1 ⎦ and ⎣ −1 1 −1 −1 ⎦ . 1 −1 −1 −1 1 1 1 −1

It is easily verified that if H1 and H2 are regular graphical Hadamard matrices with row sums ℓ1 and ℓ2 , respectively, then the Kronecker product H1 ⊗H2 is again such a matrix, whose row sum is ℓ1 ℓ2 . Starting with the above Hadamard matrices, we can make regular graphical Hadamard matrices of order m = 4t with row sum ℓ = 2t and ℓ = −2t . Many more constructions are known, for example if m = 4t4 , t ≥ 1 for ℓ = 2t2 and ℓ = −2t2 (see [16] for a survey, and [26] for some recent developments).

3. Binary codes of strongly regular graphs 3.1. Introduction Codes generated by the incidence matrix of combinatorial designs and related structures have been studied rather extensively. The best reference for this is the book by Assmus and Key [4] (see also the update [5]). Codes generated by the adjacency matrix of a graph did get less attention. For strongly regular graphs there is much analogy with designs and therefore interesting results may be expected. Concerning the dimension of these codes, that is, the p-rank of strongly regular graphs , several results are known: see [9], [33]. It has turned out that some special strongly regular graphs generate nice codes, see [23] and [38]. Here we restrict to binary codes, not only because it is the simplest case, but also since for the binary case there is a relation with regular two-graphs and Seidel switching that has already proved to be useful: see [23] and [14]. For an integral n × v matrix A we define the binary code CA of A to be the subspace of V = Fv2 generated by the rows of A (mod 2). We start with some known lemmas for symmetric integral matrices (see [9], [13] or [33]). Lemma 3.1 If A is a symmetric integral matrix with zero diagonal, then 2-rank(A) (i.e. the dimension of CA ) is even. Proof. Let A′ be a non-singular principal submatrix of A with the same 2-rank as A. Over Z, any skew symmetric matrix of odd order has determinant 0 (since det(A) = −det(A⊤ )). Reduction mod 2 shows that A′ has even order. ⊔ ⊓ Lemma 3.2 If A is a symmetric binary matrix, then diag(A) ∈ CA .   ⊥ . Then i (A)ii xi = i,j (A)ij xi xj = x⊤ Ax = 0 (mod 2), so Proof. Suppose x ∈ CA ⊥ . ⊔ ⊓ x ⊥ diag(A). Hence diag(A) ⊥ CA With these lemmas we easily find a relation between the codes CA and CA+J . Proposition 3.1 Suppose A is the adjacency matrix of a graph then CA ⊆ CA+J and the following are equivalent: (i) CA = CA+J , (ii) 1 ∈ CA , (iii) dim(CA+J ) is even.

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Proof. By Lemma 3.2, diag(A + J) = 1 ∈ CA+J , so CA+J = CA + 1 and the equivalence of (i) and (ii) follows. By Lemma 3.1 we have 2-rank(A) is even and so (i) ⇔ (iii). ⊔ ⊓ The next proposition gives a trivial but useful relation between CA and CA+I . ⊥ ⊆ CA+I with equality if Proposition 3.2 If A is a symmetric integral matrix, then CA and only if A(A + I) = 0 (mod 2). ⊥ Proof. Suppose x ∈ CA . Then Ax = 0 (mod 2), so (A + I)x = x and hence x ∈ CA+I . ⊥ Clearly A(A + I) = 0 (mod 2) reflects that CA+I ⊆ CA , which completes the proof. ⊔ ⊓

3.2. Facts from the parameters Here we present some properties of the binary codes of a strongly regular graph Γ, using only the parameters (eigenvalues) of Γ. Proposition 3.3 Suppose Γ has non-integral eigenvalues. (i) If µ is odd (i.e. v = 5 mod 8) then CA = 1⊥ and CA+I = V. ⊥ (ii) If µ is even (v = 1 mod 8) then CA = CA+I and dim(CA ) = dim(CA+I ) − 1 = 2µ (= f = g = k = (v − 1)/2). Proof. If µ is odd, Equation 2.2(ii) becomes A2 = A+I+J (mod 2), so (A+J)(A+I) = I (mod 2), hence CA+J = CA+I = V and CA = 1⊥ . Suppose µ is even. Then A2 = A ⊥ (mod 2) so CA+I = CA . The characteristic polynomial of A is given by: det(xI − A) = (x + k)(x2 + x + µ)f = xf +1 (x + 1)f (mod 2). Therefore 2-rank(A + I) ≥ v − f and 2-rank(A) ≥ v − (f + 1) = f . We know (Proposition 3.2) 2-rank(A) + 2-rank(A + I) = v, and the result follows. ⊔ ⊓ Proposition 3.4 Suppose the eigenvalues r and s of Γ are integers. (i) If k = r = s = 1 (mod 2) then CA = V, CA+I is self-orthogonal and dim(CA+I ) ≤ min{f + 1, g + 1}. (ii) If r = s = 1 (mod 2) and k is even, then CA = 1⊥ , CA+I is orthogonal to CA and dim(CA+I ) ≤ min{f + 1, g + 1}. ⊥ (iii) If r = s (mod 2) and k is even, then CA+I = CA , dim(CA ) = f ′ and ′ ′ dim(CA+I ) = v − f , where f is the multiplicity of the odd eigenvalue. ⊥ (iv) If r = s (mod 2) and k is odd, then CA = CA , dim(CA ) = f ′ + 1 and ′ dim(CA+I ) = v − f . (v) If r = s = 0 (mod 2) then k is even, CA+I = V, CA is self-orthogonal and dim(CA ) ≤ min{f + 1, g + 1} and even. Proof. (i): Equation 2.2(ii) gives A2 = I and (A + I)2 = 0 (mod 2). Over the real numbers, rank(A − rI) = v − f = g + 1, hence 2-rank(A + I) ≤ g + 1 and similarly, 2-rank(A + I) ≤ f + 1. (ii): Now A1 = 0, A2 = I + J, and (A + I)2 = J (mod 2), proving the first two claims. For the dimension bound see case (i).

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⊥ (iii): Now Equation 2.2(ii) becomes A(A + I) = 0 (mod 2), so CA+I = CA by v−f ′ f′ (x + 1) , so Proposition 3.2. The characteristic polynomial of A (mod 2) reads x dim(CA+I ) ≥ v − f ′ and dim(CA ) ≥ f ′ and, since they add up to v the result follows. (iv): Here AA = 0 (mod 2). Similar to case (iii) we get dim(CA+I ) ≥ v − f ′ − 1 and dim(CA ) ≥ f ′ + 1. Now the dimensions add up to v + 1, but f ′ is odd (from trace(A)) and v is even (since k is odd), so by Proposition 3.1 we find dim(CA ) = f ′ + 1, dim(CA+I ) = v − f ′ and dim(CA ) = v − f ′ − 1. (v): Now A2 = kJ and (A + I)2 = kJ + I (mod 2). From k + f r + gs = 0 it follows that k is even. By Lemma 3.1 dim(CA ) is even. The rest follows by similar arguments as above. ⊔ ⊓

Thus, unless r and s are both even, the dimension of CA (i.e. 2-rank(A)) follows from the parameters of Γ and similarly, dim(CA+I ) follows, unless r and s are both odd (see [9]). From the two propositions above we also see that if rs (= µ − k) is odd CA and CA+J (= CA+I ) are determined by the parameters of Γ. Similarly, CA+I and CA are determined if (r +1)(s+1) is odd. So in these cases non-isomorphic strongly regular graphs with the same parameters (of which there are many examples) generate the same (trivial) codes. 3.3. Some families and their codes 3.3.1. Triangular graphs The triangular graph T (n) is the line graph of the complete graph Kn . It follows that T (n) is a strongly regular graph with v = n(n − 1)/2, k = 2(n − 2), λ = n − 2, µ = 4, r = n − 4 and s = −2. T (n) is known to be determined by these parameters if n = 8. If N is the vertex-edge incidence matrix of Kn , then A = N ⊤ N (mod 2) is the adjacency matrix of T (n). The words of CN , CA and CA+I are characteristic vectors of subsets of the edge set of Kn , so can be interpreted as graphs on a fixed vertex set of size n. It is easily seen that CN is the n − 1 dimensional binary code consisting of all ⊥ consists of disjoint unions of Euler graphs. Note complete bipartite graphs and that CN that 1 ∈ CN . Theorem 3.1 Let Γ be the triangular graph T (n). If n is even then CA = CN ∩ 1⊥ (the Eulerian complete bipartite graphs), CA+I = V, CA = V if n = 0 (mod 4) and CA = 1⊥ if n = 2 (mod 4). ⊥ ⊥ ⊥ If n is odd then CA = CN , CA+I = CN if n = 1 (mod 4) and CA = CN ∩ 1⊥ , CA = CN (the unions of Euler graphs with an even total number of edges) if n = 3 (mod 4). Proof. Since N ⊤ N = A (mod 2), we have CA ⊂ CN . First suppose n is odd. By ⊥ . iii of Proposition 3.4, dim(CA ) = f = n − 1, hence CA = CN and CA+I = CN Proposition 3.1 gives CA = CA+I whenever (n − 1)(n − 2)/2 = dim(CA+I ) is even, that is n = 1 mod 4. If n = 3 mod 4, CA has dimension one less and is orthogonal to CA and to 1. Since 1 ∈ CA , this proves the last claim. Next take n even. By i and ii of Proposition 3.4 we find CA+I and CA . Since dim(kernel(N ⊤ )) = 1 (mod 2), dim(CA ) ≥ ⊥ ⊥ but (since n is even), 1 ∈ CN . Therefore dim(CN ) − 1 = n − 2. Clearly 1 ∈ CA ⊥ ⊥ ⊥ CA = CN + 1 and so CA = CN ∩ 1 . ⊔ ⊓ From Theorem 3.1 it follows that the codes CN and CA only have weights wi = i(n − i)

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 n (0 ≤ i ≤ n2 ). In n is odd, the number n of codewords of weight wi equals n i (for  both n CN and CA ). If n is even, CN has i codewords of weight wi for 0 ≤ i < 2 and 21 n/2 codewords of weight wn/2 . The code CA consists of the codewords from CN with even weight. 3.3.2. Lattice graphs The lattice graph L(m) is the line graph of the complete bipartite graph Km,m . It is strongly regular with parameters v = m2 , k = 2(m − 1), λ = m − 2, µ = 2, r = n − 2 and s = −2. If m = 4, L(m) is determined by these parameters. Similar to above the adjacency matrix A = M ⊤ M (mod 2) if M is the vertex-edge incidence matrix of ⊥ Km,m . The code CM consist of the edge sets of Km,m that form a union of Euler graphs. The code CM has dimension 2m − 1 and consists of disjoint unions of two bipartite graphs, one on m1 +m2 and one on (m−m1 )+(m−m2 ) vertices. Each choice of m1 , m2 (0 ≤ m1 ≤ m, 0 ≤ m2 ≤ m/2) gives codewords  m  mof  weight m1 m2 +(m−m  m 2 ).  1 )(m−m 1 m The number of these codewords equals m if m < m/2 and 2 m2 2 m1 m/2 if 1 m2 = m/2 (but note that different choices for m1 , m2 can lead to the same weight). The weight enumerators of the codes CA now follow easily from the next result. Theorem 3.2 Let Γ be the lattice graph L(m). If m is even then CA consists of the graphs from CM with m1 + m2 odd, and moreover, CA + 1 = CM and CA+I = CA = V. If m is odd then CA consists of the graphs from CM with m1 + m2 even, and moreover, ⊥ CA = CM ∩ 1⊥ , CA+I = CA and CA = CA+I ∩ 1⊥ . Proof. From M ⊤ M = A (mod 2), we deduce CA ⊆ CM and dim(CA ) ≥ dim(CM )−1 = 2m − 2. Let χ ∈ Fv2 represent a subgraph of Km,m with all vertex degrees odd (if m is ⊥ ⊥ odd, we may choose χ = 1). Then χ ∈ CA , but χ ∈ CM , hence CA = CM ∩ χ⊥ . Now all statements follow straightforwardly. ⊔ ⊓ 3.3.3. Paley graphs Suppose v = 1 (mod 4) is a prime power. The Paley graph has Fv as vertex set and two vertices are adjacent if the difference is a non-zero square in Fv . The Paley graph is an SRG(v, (v − 1)/2, (v − 1)/4 − 1, (v − 1)/4) which is isomorphic to its complement. By Propositions 3.3 and 3.4, the code CA of a Paley graph is only non-trivial if v = 1 (mod 8). Then CA and CA+I are well known as the (binary) quadratic residue codes, see for example [15] or [29] (which are usually only defined for primes v). For v = 5, 9, 13 and 17, the Paley graph is the only one with the given parameters. If v ≥ 25, other graphs with the same parameters exist. If v = 5 (mod 8) all these graphs give isomorphic (trivial) codes. If v = 25 or 41 (see Section 3.5), the known non-isomorphic graphs give non-isomorphic codes and amongst them, the codes of the Paley graphs have the largest minimum distance. We conjecture that the second part of this statement is true in general. 3.3.4. Graphs from designs and Latin squares Let D denote a 2-(n, κ, 1) design with incidence matrix N . Then A = N ⊤ N − κI is the adjacency matrix of a strongly regular graph ΓD with parameters (m2 − m(m − 1)/κ, κ(m − 1), κ2 − 2κ + m − 1, κ2 ), where m = (n − 1)(κ − 1). We have CA =

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CN N ⊆ CN if κ is even and CA+I = CN N ⊆ CN if κ is odd. If κ = 2, Γ(D) is a triangular graph and the related codes are given above. If κ = 3 D is a Steiner triple system ST S(n). A Latin square of order m (denoted by LS(m)) is an m × m matrix L with entries from {1, . . . , m} such that every entry occurs exactly once in every row and column. A Latin square can be represented by a set of m2 triples (i, j, k) indicating that entry (i, j) is equal to k. Then two triples of at most one entry in common. The Latin square graph ΓL of L is defined on the triples (the entries of L), where two triples are adjacent if they have an element in common (that is, the entries are in the same row, the same column, or have the same value). Then it easily follows that ΓL is an SRG(m2 , κ(m − 1), κ2 − 3κ + m, κ(κ − 1)). Let N be the 3m × m2 incidence of this the set of triples of a L. Then we easily have that A = N ⊤N − 3I is the adjacency matrix of ΓL , and CA+I = CN  N ⊆ CN . For ΓD and ΓL , the dimensions of CN and CA+I are known in terms of the number of sub-triple systems and quotient Latin squares, see [18], [31] and [33]. In some cases the relation between CN and CA+I is easy. Proposition 3.5 If D is an ST S(n) then (i) if n = 1 (mod 4) (i.e. m is even), then CA+I = CN and dim(CA+I ) = n; (ii) if n = 3 (mod 4) (i.e. m is odd), then dim(CA+I ) = 2dim(CN ) − n (so CA+I = CN if and only if dim(CN ) = n).

If D represents an LS(m) then dim(CN ) ≤ 3m − 2 and

(iii) if m is odd then CA+I = CN and dim(CA+I ) = 3m − 2; (iv) if m is even then dim(CA+I ) ≤ 3m − 4 with equality if and only if dim(CN ) = ⊥ 3m − 2; equality also implies that CA+I = CN ∩ CN .

Proof. The cases (i) and (iii) follow from Proposition 3.4 and the results about dimensions in (ii) and (iv) can be found in Chapter 3 of [33]. So we are left with the ⊥ last statement. We have N N ⊤ = (J3 + I3 ) ⊗ Jm (mod 2) and dim(CN ∩ CN ) ≤ ⊤ ⊤ dim(CN ) − 2-rank(N N ) = 3m − 4. Moreover, N N N = 0, so CA+I ⊥ CN and ⊥ hence CA+I ⊆ CN ∩ CN and the result follows. ⊔ ⊓ For Steiner triple systems the problem has been raised (see [38]) whether or not nonisomorphic designs always give non-isomorphic codes CN . This is true for n ≤ 15. If dim(CA+I ) < n (the ST S(n) has subsystems) then CA+I = CN . also the codes CA+I are mutually non-isomorphic. However, there exist examples of non-isomorphic strongly regular graphs with the parameters of the graph of an ST S(15), but with isomorphic codes CA+I of dimension 15 (see [24]). The binary codes of Latin squares have also been studied by Assmus [3]. he wonders if non-isomorphic Latin squares (regarded as nets of degree 3) give non-isomorphic codes CN . This is true for m ≤ 7. In particular if m = 4 the codes CN of the two Latin squares even have different dimension. However the codes CA+I of the graphs are isomorphic, because they correspond to the same 2-(16, 10, 6) design (see the end of Section 2.3). 3.4. Two-graph codes We briefly explain Seidel switching. For details we refer to [11] or [15]. Let Γ = (V, E) be a graph and let {V1 , V \ V1 } be a partition of V , then we define the result of switching

264

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Γ with respect to this partition to be the graph Γ′ = (V, E ′ ) whose edges are those edges of Γ contained in V1 or V \V1 together with the pairs {v1 , v2 }, with v1 ∈ V1 , v2 ∈ V \V1 for which {v1 , v2 } ∈ E. The graphs Γ and Γ′ are said to be switching equivalent. It is not hard to check that switching defines an equivalence relation on graphs. An equivalence class is called a two-graph. Note that, if we switch with respect to the set of neighbors Γx of a vertex x, then x becomes an isolated vertex in Γ′ . If we order the vertices in a suitable way then, in terms of the adjacency matrices A and A′ , Seidel switching comes down to ! ! A1 A12 + J A1 A12 ′ (mod 2). ,A = A= A⊤ A2 A⊤ 12 + J 12 A2 Suppose we switch with respect to a subset V1 of V with characteristic vector χ. Then we have CA + 1 + χ = CA′ + 1 + χ . Let us not worry about 1 and look at the codes CA+J = CA + 1 and CA′ +J . It is clear that if χ ∈ CA+J then CA′ +J ⊆ CA+J . Suppose Γ and Γ′ both have an isolated vertex (not the same one) then χ is in CA+J and CA′ +J , hence CA+J = CA′ +J . So this code is independent of the isolated vertex and we will call it the two-graph code. Note that 1 ∈ CA (because of the isolated vertex), so dim(CA+J ) = dim(CA ) + 1 is odd. Assume Γ is an SRG(v, k, λ, µ) with k = 2µ (or equivalently, k = −2rs). Extend Γ ˜ (i.e. Γ\{x} ˜ ˜ to Γ ˜ ′ , such that another with an isolated vertex x to Γ = Γ). If we switch in Γ ′ ′ ˜ \ {y} is again a SRG(v, k, λ, µ), vertex y becomes isolated, then it follows that Γ = Γ ˜ is called a but not necessarely isomorphic to Γ. In this case the switching class of Γ ˜ with respect to x (and y). regular two-graph and Γ (and Γ′ ) is the descendant of Γ Clearly, the code CA of a descendant is the shortened code of the corresponding twograph code. Regular two-graphs can produce interesting two-graph codes. For example the Paley graph is the descendant of a regular two-graph and the corresponding twograph code is the extended quadratic residue code. For other interesting two-graph codes, ˜ can be switched into a regular graph Γ ˜ ′ , then it follows that Γ ˜′ see [14], [22] and [23]. If Γ is strongly regular with the same r and s as Γ, but with two possibilities for the valency: Either k = −2rs − r or k = −2rs − s (so r and s need to be integral). On the other hand, a strongly regular graph with degree −2rs − r or −2rs − s is in the switching class of a regular two-graph (so isolating a vertex yields a strongly regular graph with k = −2rs). For example the Shrikhande graph, L(4) and the complement of the Clebsch graph are switching equivalent. We observed already that these three graphs generate the same (6-dimensional) code. By isolating a vertex we get T (6) and the two-graph code is a 5-dimensional subspace of the L(4) code. The shortened code (with respect to any vertex) is the 4-dimensional code of T (6). Theorem 3.3 Suppose Ω is a regular two-graph with eigenvalues r and s and two-graph code C. Suppose Γ is a k-regular graph in Ω (so Γ is strongly regular) and let ∆ be the graph in Ω with a given vertex x isolated (so switching in Γ with respect to the neighbors Γx of x gives ∆). Let A and B be the adjacency matrices of Γ and ∆ respectively, and let χ denote the characteristic vector of the switching set Γx . Then either

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⎫ ⎪ ⎪ ⎬

⎧ dim CA = dim CB = dim C − 1 dim CA − 2 = dim CB = dim C − 1 ⎪ ⎪ ⎨ 1 ∈ CA 1 ∈ CA or χ ∈ CB χ ∈ CB ⎪ ⎪ ⎪ ⎭ ⎪ ⎩ CA+J = CA = CB + 1 + χ = C + χ. CA+J = CA + 1 = CB + 1 = C

If k is even and r + s is odd, we are in the first case. If k = 2 mod 4 and r + s is even, or k is odd, we are in the second case. Proof. The results follow from the fact that

CA + 1 + χ = CB + 1 + χ, CA + 1 = CA+J , C = CB + 1, and that dim CA and dim CB are even. Clearly 1 ∈ CB , 1 ∈ CA+J and χ ∈ CA . If 1 ∈ CA then CA = CA+J and CA = CB + 1 + χ, so CB is a proper subspace of CA and hence dim CA = dim CB + 2 and χ ∈ CB . On the other hand, if 1 ∈ CA , then χ must be a codeword of CB and dim CA = dim CB . Furthermore, CA+J =CA +1 =CB +1 = C. If k is even and r + s is odd, then µ = k + rs is even and λ = µ + r + s is odd. Now the rows of B corresponding to Γx add up to the characteristic vector χ of Γx . So χ ∈ CB and hence we are in the first case. It is clear that 1 ∈ CA if k is odd. Suppose k = 2 mod 4 and r + s is even. Then r and s are both even (since −k = 2rs + s or 2rs + r). Let B ′ be the adjacency matrix of the descendant ∆′ = ∆ \ {x}. Then CB ′ is self-orthogonal by 3.4.v. Moreover, the degree of ∆′ is 2rs, which is divisible by 4, and hence all weights in CB ′ and CB are ⊔ ⊓ divisible by 4. Therefore χ ∈ CB , so we are in the second case. For example, the last statement implies that 1 ∈ CA for an SRG(36, 14, 4, 6). If k is even and r + s is odd then C = CA+J . So, in this case, non-isomorphic switching equivalent strongly regular graphs give isomorphic codes of the form CA+J . Examples are given by the switching equivalent SRG(26, 10, 3, 4)’s (see the next section). It is clear that if two two-graph codes are isomorphic then so are the codes of corresponding descendants. And vice versa, two descendants Γ1 and Γ2 with isomorphic codes CA1 = CA2 give isomorphic two-graph codes. Among the regular two-graphs on 36 vertices (r = 2, s = −4) there exist several non-isomorphic ones with isomorphic two-graph codes, therefore we also have non-isomorphic SRG(35, 16, 6, 8)’s with isomorphic codes CA (see [24]). 3.5. Small cases In Table 1 we give the parameters of all primitive strongly regular graphs on at most 40 vertices (up to taking complements). We indicate how many non-isomorphic graphs there exist with the given parameters and, if k = 2µ we give the number of corresponding non-isomorphic regular two-graphs. In the previous sections we have obtained the codes of several of these graphs. For the other parameters we refer to [24]. The mentioned paper also contains the weight enumerators of most of the codes. Here we restrict to the strongly regular graphs on 25 and 26 vertices, and the related regular two-graphs on 26 vertices. There are exactly four non-isomorphic regular two-graphs on 26 vertices with eigenvalues 2 and −3. Together they have fifteen SRG(25, 12, 5, 6)’s (two from LS(5)′ s one of which is the Paley graph) as a descendant and ten SRG(26, 15, 8, 9)’s

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266 no.

(v, k, λ, µ)

a name

#graphs

#two-graphs

dim(CA )

dim(CA )

1

(5,2,0,1)

2 3

(9,4,1,2) (10,3,0,1)

pentagon (Paley)

1

1

4

4

L(3) Petersen (T (5))

1 1

1

4 6

4 4

4 5 6

(13,6,2,3)

Paley

1

1

12

12

(15,6,1,3) (16,5,0,2)

T (6) Clebsch

1 1

1

14 16

4 6

7 8

(16,6,2,2) (17,8,3,4)

L(4) Paley

2 1

6 8

16 8

1

9

(21,10,3,6)

T (7)

1

14

6

10

(25,8,3,2)

L(5)

1

8

16

11 12

(25,12,5,6) (26,10,3,4)

LS(5) ST S(13)

13 14

(27,10,1,5) (28,12,6,4)

Schläfli T (8)

15 16 17

(29,14,6,7) (35,16,6,8) (36,10,4,2)

Paley ST S(15) L(6)

18

(36,14,4,6)

HJsub

19 20 21

(36,14,7,4) (36,15,6,6) (37,18,8,9)

T (9) LS(6) Paley

22

(40,12,2,4)

GQ(3, 3)

15 10

4

12 12

12 14

1 4

1

26 6, 8

6 28

41 3854 1

6 227

28 6,..,14 10

28 34 36

180

8,..,14

36

1 32548 ≥ 6760

8 36 36

27 6,..,16 36

10,..,16

40

≥ 191

28

Table 1. Primitive strongly regular graphs on fewer than 45 vertices

name

dim

0

s1

12

1

s2 s3

12 12

1 1

s4 s5

12 12

s6 s7

4

6

8

10

12

14

16

18

20

50

225

880

1225

1050

550

100

15

10 12

37 43

279 279

712 696

1343 1331

1140 1152

432 448

124 124

15 9

1 1

4 4

54 66

213 225

868 832

1237 1201

1062 1098

546 582

96 84

15 3

12 12

1 1

3

51 54

213 225

876 864

1243 1225

1056 1074

538 550

96 84

18 15

1 4

s8 s9

12 12

1 1

6 8

32 38

291 291

728 712

1331 1319

1122 1134

436 452

132 132

15 9

2

s10 s11 s12 s13 s14

12 12 12 12 12

1 1 1 1 1

7 5 7 6 7

39 41 35 36 35

295 303 291 295 291

708 700 720 716 720

1313 1301 1325 1319 1325

1140 1152 1128 1134 1128

456 464 444 448 444

128 120 132 128 132

8 6 12 11 12

1 3 1 2 1

s15

12

1

6

44

303

692

1295

1158

472

120

3

2

Table 2. Weight enumerators of the codes of the SRG(25, 12, 5, 6)’s.

22 3 1

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267

0

4

5

6

7

8

9

10

11

12

name

dim

26

22

21

20

19

18

17

16

15

14

13

ls11

14

1

10

65

190

325

740

1430

1826

2275

st11

14

1

13

52

130

403

884

1144

1950

2483

2264

st12

14

1

13

24

52

130

403

788

1144

1950

2483

2408

ls21

14

1

4

14

69

190

309

724

1414

1826

2299

2684

ls22

14

1

4

10

69

190

309

740

1414

1826

2299

2660

st21

14

1

8

26

47

130

423

780

1164

1950

2453

2420

st22 st23 st24

14 14 14

1 1 1

8 8 8

22 26 10

47 47 47

130 130 130

423 423 423

796 780 844

1164 1164 1164

1950 1950 1950

2453 2453 2453

2396 2420 2324

st25

14

1

8

22

47

130

423

796

1164

1950

2453

2396

2660

Table 3. Weight enumerators of the codes of the SRG(26, 15, 8, 9)’s.

(two from ST S(13)′ s) in the switching class, see [32] and [2]. The corresponding codes of the form CA have been generated and the weight enumerators are given in Table 2 and Table 3 (keeping the names and order from [32]; the lines give the partition into the four switching-equivalence classes (two-graphs)). All codes are non-isomorphic. In most cases this follows from the weight enumerator, but in some cases more information is needed; see [24]. It follows that also the four two-graph codes are non-isomorphic and by Theorem 3.3 we have that the ten graphs on 26 vertices give rise to just four non-isomorphic codes of the ⊥ + 1). In other words, by deleting the words of odd weight, the ten form CA+J (= CA codes of length 26 collapse to the four two-graph codes.

4. Divisible Design Graphs In this section we generalize the concept of a (v, k, λ)-graph, and introduce graphs with the property that the neighborhood design is a divisible design. Definition 4.1 A k-regular graph is a divisible design graph (DDG for short) if the vertex set can be partitioned into m classes of size n, such that two distinct vertices from the same class have exactly λ1 common neighbors, and two vertices from different classes have exactly λ2 common neighbors. ✉ ✚❩ ✚ ✓❙ ❩ ✉ ❙❳❩ ✚✘✓✘ ❩ ❳❩ ✚❳ ✉ ✘✘✓ ❳✉ ✚ ❙✏ P ✚ ❩ ❩ ✚ ✉ ✉✏ P ✓ ❙ ❇❇ ❇ ✂ ✓ ✂✂ ❙ ❇ ✂ ✂ ❇ ❙ ✉ ✉ ✂✓ ✂ ✓ ✦ ❇ ❙❇❛✦ ✜ ❛ ❭❭✂✉ ❇✉ ✜ ✦ ❛❛ ✦ Figure 2. A proper divisible design graph

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For example the graph of Figure 2 (which is the strong product of K2 and C5 ) is a DDG with parameters (v, k, λ1 , λ2 , m, n) = (10, 5, 4, 2, 5, 2). Note that a DDG with m = 1, n = 1, or λ1 = λ2 is a (v, k, λ) graph. If this is the case, we call the DDG improper, otherwise it is called proper. The definition of a divisible design (often also called group divisible design) varies. We take the definition given in Bose [8]. Definition 4.2 An incidence structure with constant block size k is a (group) divisible design whenever the point set can be partitioned into m classes of size n, such that two points from one class occur together in λ1 blocks, and two points from different classes occur together in exactly λ2 blocks. A divisible design D is said to have the dual property if the dual of D (that is, the design with the transposed incidence matrix) is again a divisible design with the same parameters as D. From the definition of a DDG it is clear that the neighborhood design of a DDG is a divisible design D with the dual property. Conversely, a divisible design with a polarity with no absolute points is the neighborhood design of a DDG. A DDG is closely related to a strongly regular graph. It follows easily that a proper DDG is strongly regular if and only if the graph or the complement is mKn , the disjoint union of m complete graphs of size n. Deza graphs (see [19]) are k-regular graphs which are not strongly regular, and where the number of common neighbors of two distinct vertices takes just two values. So proper DDGs, which are not isomorphic to mKn or the complement, are Deza graphs. 4.1. Eigenvalues With the identity matrix Im of order m, and the n × n all-ones matrix Jn we define K = K(m,n) = Im ⊗ Jn = diag(Jn , . . . , Jn ). Then we easily have that a graph Γ is a DDG with parameters (v, k, λ1 , λ2 , m, n) if and only if Γ has an adjacency matrix A that satisfies: A2 = kIv + λ1 (K(m,n) − Iv ) + λ2 (Jv − K(m,n) ).

(1)

Clearly v = mn, and taking row sums on both sides of Equation 1 yields k 2 = k + λ1 (n − 1) + λ2 n(m − 1). So we are left with at most four independent parameters. Some obvious conditions are 1 ≤ k ≤ v − 1, 0 ≤ λ1 ≤ k, 0 ≤ λ2 ≤ k − 1. From Equation (1) strong information on the eigenvalues of A can be obtained. (Throughout we write eigenvalue multiplicities as exponents.) Lemma 4.1 The eigenvalues of the adjacency matrix of a DDG with parameters (v, k, λ1 , λ2 , m, n) are 9 g1  9 g2 .  f1  9  f2  9 , , − k 2 − λ2 v k − λ1 k 2 − λ2 v , − k − λ1 , k1 ,

where f1 + f2 = m(n − 1), g1 + g2 = m − 1 and f1 , f2 , g1 , g2 ≥ 0.

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Proof. The eigenvalues of K(m,n) are {0m(n−1) , nm }. Because Iv , Jv and K(m,n) commute it is straightforward to compute the eigenvalues of A2 from equation (1). They are {(k 2 )1 , (k − λ1 )m(n−1) , (k 2 − λ2 v)m−1 }, and must be the squares of the eigenvalues of A.

⊔ ⊓

Some of the multiplicities may be 0, and some values may coincide. In general, the multiplicities f1 , f2 , g1 and g2 are not determined by the parameters, but if we know one, we know them all because f1 + f2 = m(n − 1), g1 + g2 = m − 1, and 9 9 trace A = 0 = k + (f1 − f2 ) k − λ1 + (g1 − g2 ) k 2 − λ2 v. (2) This equation leads to the following result.

Theorem 4.3 Consider a proper DDG with parameters (v, k, λ1 , λ2 , m, n), and eigenvalue multiplicities (f1 , f2 , g1 , g2 ). a. k − λ1 or k 2 − λ2 v is a nonzero square. b. If k − λ1 is not a square, then f1 = f2 = m(n − 1)/2. c. If k 2 − λ2 v is not a square, then g1 = g2 = (m − 1)/2. Proof. If one of k−λ1 and k 2 −λ2 v equals 0, then Equation (2) gives that the other one is a nonzero square. If k−λ1 and k 2 −λ2 v are both non-squares, it follows straightforwardly that the square-free parts of these numbers are equal non-squares, hence Equation (2) has no solution. The second and third statement are obvious consequences of Equation (2). ⊔ ⊓ If k − λ1 , or k 2 − λ2 v is not a square, the multiplicities (f1 , f2 , g1 , g2 ) can be computed from the parameters. The outcome must be a set of nonnegative integers. This gives a condition on the parameters, which is often referred to as the rationality condition. Only if k − λ1 and k 2 − λ2 v are both squares (that is, all eigenvalues of A are integers), the parameters do not determine the spectrum. Then 0 ≤ g1 ≤ m − 1, so there are at most m possibilities for the set of multiplicities. 4.2. The quotient matrix The vertex partition from the definition of a DDG gives a partition (which will be called the canonical partition) of the adjacency matrix ⎡ ⎤ A1,1 · · · A1,m ⎢ ⎥ A = ⎣ ... . . . ... ⎦ . Am,1 · · · Am,m

We shall see that the canonical partition is equitable, which means that each block Aij has constant row (and column) sum. For this, we introduce the v × m matrix S, whose columns are the characteristic vectors of the partition classes. Then S satisfies S = Im ⊗ 1n , S ⊤ S = nIm , SS ⊤ = K(m,n) ,

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270

where 1n denotes the all-ones vector with n entries. Next we define R = n1 S ⊤ AS, which means that each entry rij of R is the average row sum of Aij . We will call R the quotient matrix of A. Theorem 4.4 The canonical partition of the adjacency matrix of a proper DDG is equitable, and the quotient matrix R satisfies R2 = RR⊤ = (k 2 − λ2 v)Im + λ2 nJm . The eigenvalues of R are  9 g1  9  g2  k1 , , − k 2 − λ2 v . k 2 − λ2 v

Proof. Equation (1) gives (λ1 − λ2 )K(m,n) = A2 − λ2 Jv − (k − λ1 )Iv . Clearly A commutes with the right hand side of this equation and therefore with K(m,n) . Thus ASS ⊤ = SS ⊤ A. Using this we find: SR =

1 ⊤ n SS AS

=

1 ⊤ n ASS S

= AS,

which reflects that the partition is equitable. Similarly, R2 =

1 ⊤ ⊤ n2 S ASS AS

=

1 ⊤ 2 nS A S

= (k 2 − λ2 v)Im + λ2 nJm ,

where in the last step we used k 2 =√k+λ1 (n−1)+λ2 n(m−1). From the formula for R2 it follows that R has eigenvalues ± k 2 − λ2 v, whose multiplicities add up to m−1. If v is an eigenvector of R, then Sv is an√eigenvector of A for the same eigenvalue. Therefore the multiplicity of an eigenvalue ± k 2 − λ2 v of R is at most equal to the multiplicity of the same eigenvalue of A. This implies that the multiplicities are the same. ⊔ ⊓ The above lemma can easily be generalized to divisible designs with the dual property. This more general version of the lemma is due to Bose [8] (who gave a much longer proof). If one wants to construct a DDG with a given set of parameters, one first tries to construct a feasible quotient matrix. For this the following straightforward properties of R can be helpful: Proposition 4.1 The quotient matrix R of a DDG satisfies  i (R)i,j = k for j = 1, . . . , m,  2 2 2 i,j (R)i,j = trace(R ) = mk − (m − 1)λ2 v, √ 0 ≤ trace(R) = k + (g1 − g2 ) k 2 − λ2 v ≤ m(n − 1). In some cases these conditions lead to nonexistence or limited possibilities for R. Proposition 4.2 If m = 3 and k 2 − λ2 v is not a square, then the following system of equations has an integral solution. X + Y + Z = k, X + Y 2 + Z 2 = k 2 − 2λ2 v/3, 3 X + Y 3 + Z 3 = 3XY Z + k(k 2 − λ2 v) . 2

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Proof. The quotient matrix R is a symmetric 3 × 3 matrix with all row and column sums equal to k and, since k 2 − vλ2 is not a square, also trace(R) = k. This implies ⎡ ⎤ XY Z R = ⎣Y Z X⎦ , ZXY

so trace(R2 ) = 3(X 2 + Y 2 + Z 2 ) = k 2 + 2(k 2 − λ2 v). The third equation comes from det R = −k(k 2 − λ2 v). ⊔ ⊓

For example a DDG with parameters (21, 12, 8, 6, 3, 7) does not exist because X 2 + Y 2 +Z 2 = 60 has no integral solution. Note that Construction 4.11 gives infinitely many DDGs that satisfy the condition of the above proposition. Proposition 4.3 There exists no DDG for the parameter sets (14, 10, 6, 7, 7, 2), and (20, 11, 2, 6, 10, 2). Proof. In both cases n = 2, so trace R ≤ m. For the first parameter set this gives a contradiction, because trace R = k = 10 and m = 7. For the second parameter set, Theorem 4.5 implies that R = J + P for some symmetric permutation matrix P . Therefore trace R = 10, P has zero diagonal, and the spectrum of R is {11, 14 , −15 }. This implies that the adjacency matrix has eigenvalues 11, 3f1 , −3f2 , 14 and −15 where f1 + f2 = 10. This is impossible. ⊔ ⊓ The following result is essentially due to Bose [8] (though his formulation is different). Theorem 4.5 Consider a DDG with parameters (v, k, λ1 , λ2 , m, n). Write k = mt + k0 for some integers t and k0 with 0 ≤ k0 ≤ m − 1. Then the entries of R take exactly one, or two consecutive values if and only if k02 − mk0 − k 2 + km + λ1 m(n − 1) = 0 . If this is the case then R = tJ + N , where N is the incidence matrix of a (possibly degenerate) (m, k0 , λ0 ) design with a polarity. Proof. If each entry of R equals t or t + 1, then in each row k0 entries are equal to t + 1 and m − k0 entries are equal to t (because the row sums of R are k). Therefore, mk0 (t + 1)2 + mt2 (m − k0 ) = trace(R2 ) = mk 2 + (m − 1)λ2 v, which leads to k02 −mk0 −k 2 +km+λ1 m(n−1) = 0. Conversely, if the equation holds, then a matrix R with k0 entries t + 1 in each row, and all other entries equal to t satisfies the conditions of Equation 4.1. Moreover, any other solution to these equations has the same properties. (Indeed changing some entries to integer values different from t and t + 1, such that the sum of the entries remains the same, increases the sum of the squares of the entries). Suppose R = tJ + N for some incidence structure N , then N = N ⊤ , and Theorem 4.4 implies that N 2 ∈ J, I, therefore N is the incidence matrix of a (m, k0 , λ) design. ⊔ ⊓ Note that the √ number of absolute points of the polarity equals trace N = trace R − mt = k + (g1 − g2 ) k 2 − λ2 v − mt, which is equal to k − mt = k0 if k 2 − λ2 v is not a square.

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4.3. Constructions In this section we present some constructions of DDGs. 4.3.1. (v, k, λ) graphs and designs We recall that the incidence graph of a design with incidence matrix N is the bipartite graph with adjacency matrix ! O N . N⊤ O Construction 4.6 The incidence graph of an (n, k, λ1 ) design with 1 < k ≤ n is a proper DDG with λ2 = 0. Construction 4.7 The disconnected graph for which each component is an (n, k, λ1 ) graph (1 < k < n), or the incidence graph of an (n, k, λ1 ) design (1 < k ≤ n), is a proper DDG with λ2 = 0. Proposition 4.4 For a proper DDG Γ the following are equivalent. a. Γ comes from Construction 4.6, or 4.7. b. Γ is bipartite or disconnected. c. λ2 = 0. Proof. It is clear that a bipartite or disconnected DDG has λ2 = 0. Assume Γ is a DDG with λ2 = 0. Then in every block row of the canonical partition of the adjacency matrix there is exactly one nonzero block (otherwise the neighborhood of a vertex contains vertices in different blocks which contradicts λ2 = 0), and each nonzero block is the incidence matrix of a (n, k, λ1 ) design. If such a block is on the diagonal it is the adjacency matrix of a (n, k, λ1 ) graph with 1 < k < n. If it is not on the diagonal the transposed block is on the transposed position, and together they make the bipartite incidence graph ⊔ ⊓ of a (n, k, λ1 ) design with 1 < k ≤ n. Construction 4.8 If A′ is the adjacency matrix of a (m, k′ , λ′ ) graph (1 ≤ k ′ < m), then A′ ⊗ Jn is the adjacency matrix of a proper DDG with k = λ1 = nk ′ , λ2 = nλ′ . Proposition 4.5 For a proper DDG Γ the following are equivalent. a. Γ comes from Construction 4.8. b. The adjacency matrix of Γ can be written as A′ ⊗ Jn for some m × m matrix A′ . c. λ1 = k. Proof. The only nontrivial claim is that c implies a. Assume Γ is a DDG with k = λ1 . Then any two rows of the adjacency matrix belonging to the same class are identical. Since the blocks have constant row and column sum this implies that all blocks have only ones, or only zeros. Therefore the adjacency matrix has the form A′ ⊗ Jn , where A′ is a symmetric (0, 1)-matrix with zero diagonal and row sum k/n. Moreover, any two distinct rows of A′ have inner product λ2 /n. Therefore A′ is the adjacency matrix of a ⊔ ⊓ (m, k ′ , λ′ ) graph.

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Construction 4.9 Let A1 , . . . , Am (m ≥ 2) be the adjacency matrices of m (n, k ′ , λ′ ) graphs with 0 ≤ k ′ ≤ n − 2. Then A = J − K + diag(A1 , . . . , Am ) is the adjacency matrix of a proper DDG with k = k ′ + n(m − 1), λ1 = λ′ + n(m − 1)), λ2 = 2k − v. Proposition 4.6 For a proper DDG Γ the following are equivalent. a. Γ comes from Construction 4.9. b. The complement of Γ is disconnected. c. λ2 = 2k − v. Proof. Let x and y be two vertices of Γ. Simple counting gives that the number of common neighbors is at most 2k − v, and equality implies that x and y are adjacent. So, if λ2 = 2k − v, then two vertices from different classes are adjacent, and hence the complement is disconnected. Conversely, suppose Γ is a DDG with disconnected complement G (say). Let x and y be vertices in different components of G. Then x and y have no common neighbors in G, and hence x and y are adjacent vertices in Γ with 2k − v common neighbors. Therefore λ2 = 2k − v, and all vertices from different classes are adjacent. Finally, equivalence of a and b is straightforward. ⊔ ⊓ Note that in the above constructions the used (v, k, λ) graphs and designs may be degenerate. This means that the above constructions include the k-regular complete bipartite graph (k ≥ 2), the (k + 1)-regular complete bipartite graph minus a perfect matching (k ≥ 2), the disjoint union of m complete graphs Kn (m ≥ 2, n ≥ 3), the complete m-partite graph with parts of size n (m ≥ 2, n ≥ 2), and the complete m-partite graphs with parts of size n extended with a perfect matching of the complement (m ≥ 2, n ≥ 4, n even). So these DDGs exist in abundance, and we’ll call them trivial. 4.3.2. Hadamard matrices Construction 4.10 Consider a regular graphical Hadamard matrix H of order m ≥ 4 √ and row sum ℓ = ± m. Let n ≥ 2. Replace each entry with value −1 by Jn − In , and each +1 by In , then we obtain the adjacency matrix of a DDG with parameters (mn, n(m − ℓ)/2 + ℓ, (n − 2)(m − ℓ)/2, n(m − 2ℓ)/4 + ℓ, m, n). In terms of the adjacency matrix the construction becomes: H ⊗ In + 21 (J − H) ⊗ Jn . Using this, it is straightforward to check that Equation 1 is satisfied. We recall (see Section 2.3) the two regular graphical Hadamard matrices of order 4: ⎤ ⎤ ⎡ ⎡ −1 −1 −1 1 −1 1 1 1 ⎢ −1 −1 1 −1 ⎥ ⎢ 1 −1 1 1 ⎥ ⎥ ⎥ ⎢ ⎢ ⎣ 1 1 −1 1 ⎦ and ⎣ −1 1 −1 −1 ⎦ . 1 −1 −1 −1 1 1 1 −1

For the first one, the DDG is the 4 × n grid, that is, the line graph of K4,n . The second one gives DDGs with parameters (4n, 3n − 2, 3n − 6, 2n − 2, 4, n); for n = 2 this is the complement of the cube. The DDGs of Construction 4.10 are improper whenever λ1 = λ2 , which is the case if and only if n = 4.

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Construction 4.11 Consider a regular graphical Hadamard matrix H of order ℓ2 ≥ 4 with diagonal entries −1 and row sum ℓ. The graph with adjacency matrix ⎡ ⎤ M N O A = ⎣ N O M ⎦ , where O M N 1 J +H J +H M= 2 J +H J +H

!

1 J +H J −H , and N = 2 J −H J +H

!

,

is a DDG with parameters (6ℓ2 , 2ℓ2 + ℓ, ℓ2 + ℓ, (ℓ2 + ℓ)/2, 3, 2ℓ2 ). For the two Hadamard matrices presented above, this leads to DDGs with parameters (24, 10, 6, 3, 3, 8) and (24, 6, 2, 1, 3, 8), respectively. 4.3.3. Divisible designs Here we examine known constructions of divisible designs that admit a symmetric incidence matrix with zero diagonal, and therefore correspond to DDGs. Clearly, we can restrict ourselves to divisible designs with the dual property. Many constructions for these kind of designs come from divisible difference sets. Such a construction uses a group G of order v = mn, together with a subset of G of order k, called the base block. The blocks of the design are the images of the base block under the group operation. Thus we obtain v blocks of size k (blocks may be repeated). This construction gives a divisible design if the group G has a normal subgroup N of order n and the base block is a so called divisible difference set relative to N . It follows from the construction that such a divisible design has the dual property. Moreover, one can order the points and blocks such that the incidence matrix becomes symmetric, and it is also easy to find an ordering that gives a zero diagonal. The problem is to find an ordering that simultaneously provides a symmetric matrix and a zero diagonal. Such an ordering is not always possible. For having a symmetric incidence matrix with zero diagonal, the divisible difference set should be reversible (or equivalently, it must have a strong multiplier −1). Several reversible relative difference sets are known. For example, for the group G = C5 × S2 = {1, a, a2 , a3 , a4 } × {1, b} the base block {(1, b), (a, 1), (a, b), (a4 , 1), (a4 , b)} is a reversible difference set relative to N = S2 , and hence gives a DDG. This DDG is the one given in Figure 2. In fact, several of the examples constructed so far can also be made with a reversible divisible difference set. These include all trivial examples and some of the ones from Construction 4.10. For more examples and information on reversible difference sets we refer to [1]. Another useful result on divisible designs is the construction and characterization of divisible designs with k − λ1 = 1 given in [20]. We recall that the strong product of two graphs with adjacency matrices A and B, is the graph with adjacency matrix (A + I) ⊗ (B + I) − I. Construction 4.12 Let Γ′ be a strongly regular graph with parameters (m, k ′ , λ, λ + 1). Then the strong product of K2 with Γ′ is a DDG with n = 2, λ1 = k − 1 = 2k ′ and λ2 = 2λ + 2.

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Checking the correctness of the construction is straightforward. There exist infinitely many strongly regular graphs with the required property. For example the Paley graphs. But there are infinitely many others. It easily follows that the complement of a strongly regular graph with µ − λ = 1 has the same property. Thus we can get two DDGs from one strongly regular graph with µ − λ = 1, unless the strongly regular graph is isomorphic to the complement (which is the case for the Paley graphs). For example the Petersen graph and its complement lead to DDGs with parameters (v, k, λ1 , λ2 , m, n) = (20, 7, 6, 2, 10, 2) and (20, 13, 12, 8, 10, 2), respectively. The pentagon, which is a strongly regular graph with parameters (5, 2, 0, 1), leads once more to the example of Figure 2. In fact, several graphs coming from Construction 4.12 can also be constructed by use of a reversible divisible difference set. This includes all Paley graphs. Theorem 4.13 Let Γ be a nontrivial proper DDG, then Γ comes from Construction 4.12 if and only if k − λ1 = 1. Proof. Assume Γ is a DDG with k − λ1 = 1. According to [20] the neighborhood design D, or its complement has incidence matrix N = (A ⊗ Jn ) + Iv , where one of the following holds: (i) J − 2A is the core of s skew-symmetric Hadamard matrix (this means that A + A⊤ = J − I, and 4AA⊤ = (v + 1)I + (v − 3)J). (ii) n = 2, and A is the adjacency matrix of a strongly regular graph with µ − λ = 1, or (iii) A = O, or A = J −I. Case iii and its complement correspond to trivial DDGs. Case ii corresponds to Construction 4.12 (note that N has no zero diagonal, but interchanging the two rows in each class gives N the required property). Also the complement of Case ii corresponds to Construction 4.12. Indeed, Jv − N = Jv − (A ⊗ J2 ) − Iv = (Jm − A) ⊗ J2 − Iv , where A, and therefore also Jm − A − Im is the adjacency matrix of a strongly regular graph with µ − λ = 1. Finally we will show that Case i is not possible for a DDG. Suppose P N = P (A ⊗ J) + P , or P (J − N ) is symmetric with zero diagonal for some permutation matrix P , then P is symmetric and preserves the block structure. The quotient matrix Q of P is a symmetric permutation matrix such that QA is symmetric with zero diagonal. We have A + A⊤ = J − I, so J − Q = AQ + A⊤ Q = AQ + QA, and therefore trace(J − Q) = 2 trace(QA) = 0, so Q = I, a contradiction. ⊔ ⊓ 4.3.4. Partial complements The complement of a DDG is almost never a DDG again. If the partition classes are the same, then only the complete multipartite graph and its complement have this property. The cube (which is a bipartite DDG with two classes) and its complement (which is a DDG with four classes) is an example where the canonical partitions differ. However, if we only take the complement of the off-diagonal blocks it is more often the case that we get a DDG again. We call this the partial complement of the DDG. We have seen one such example in Construction 4.12, where the partial complement can be constructed in the same way, and hence produces no new examples. The following idea however can give new examples. Proposition 4.7 The partial complement of a proper DDG Γ is again a DDG if one of the following holds: a. The quotient matrix R equals t(J − I) for some t ∈ {1, . . . , n − 1}. b. m = 2.

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Proof. We use Equation 1. In Case a, the partial complement has adjacency matrix : = J − K − A. In Section 4.2 we saw that AK = KA = ASS ⊤ = SRS ⊤ . Since A :2 ∈ Span {I, J, K}, and A : R = t(J − I) this implies AK ∈ Span {J, K}. Therefore A represents a DDG. In Case b, the vertices can be ordered such that the partial complement has adjacency ma: = J −K +DAD, where D = diag(1, . . . , 1, −1, . . . , −1). The quotient matrix R trix A is a symmetric 2 × 2 matrix with constant row sum, hence R ∈ Span {I2 , J2 }, and therefore AK = SRS ⊤ ∈ Span {K2,n , Jv }, and also DADK = DAK ∈ Span {K2,n , Jv }. :2 ∈ Span {I, J, K}, Moreover, (DAD)2 = DA2 D ∈ Span {Iv , Jv , K2,n }, and hence A which proves our claim. ⊔ ⊓ Taking partial complements often gives improper DDGs. Conversely, the arguments also work if Γ is an improper DDG (that is, Γ is a (v, k, λ) graph), provided Γ admits a nontrivial equitable partition that satisfies a or b. An equitable partition of a (v, k, λ) graph that satisfies a is a so called Hoffman coloring (see [25]). Note that the diagonal blocks are zero, so the partition corresponds to a vertex coloring. Thus we have:

Construction 4.14 Let Γ be a (v, k, λ) graph. If Γ has a Hoffman coloring, or an equitable partition into two parts of equal size, then the partial complement is a DDG. Also this construction can give improper DDGs, but in many cases the DDG is proper. For example there exists a strongly regular graph Γ with parameters (v, k, λ, µ) = (40, 12, 2, 4) with a so called spread, which is a partition of the vertex set into cliques of size 4 (see [25]). The complement of Γ is a (40, 27, 18) graph, and the spread of Γ is a Hoffman coloring in the complement. The partial complement is Γ with the edges of the cliques of the spread removed. This gives a DDG with parameters (40, 9, 0, 2, 10, 4). By taking the union of five classes in this Hoffman coloring, we obtain an equitable partition into two parts of size 20. The partial complement with respect to this partition gives a DDG with parameters (40, 17, 8, 6, 2, 20).

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-278

Finding error-correcting codes using computers Clement LAM Department of Computer Science and Software Engineering, Concordia University, Montreal, Quebec, Canada Abstract. The theory of error-correcting codes is a vast and fast-moving area with many open problems. The objective of this paper is to survey where the current boundaries of knowledge are in a few selected areas, by listing the smallest unsolved cases. Our hope is that these lists will motivate further computational to move these boundaries. Keywords. error-correcting code, self-dual

Introduction We start with the basic definitions. For a detailed introduction to the subject, please consult the standard references, such as [2,11,17]. A code C of length n over an alphabet Q of size q is a subset C ⊆ Qn , where n Q is the set of all n-tuples with entries from Q. We will assume that 0 ∈ Q. A code is binary if q = 2, ternary if q = 3, and quaternary if q = 4. For q > 4, we will just call them q-ary codes. The elements of a code C are called codewords. If x, y ∈ Qn , then the Hamming distance d(x, y) is defined by  d(x, y) = |{i|xi = yi }|, where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). The distance distribution of a code C is given by the sequence (Ai )ni=0 where Ai =

1 |{(x, y)|x, y ∈ C, d(x, y) = i}|. |C|

The Hamming weight of a codeword x is w(x) = d(x, ¯0), where ¯0 = (0, . . . , 0). Beside the Hamming distance, there is also the Lee distance. If Q = {0, . . . , q− 1}, the Lee weight of i is defined by wL (i) = min{i, q − i}.

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The Lee weight of a codeword x = (x1 , . . . , xn ) is wL (x) =

n 

wL (xi ),

i=1

and the Lee distance of codewords x and y is defined by dL (x, y) = wL (x − y). In this paper, the unqualified words weight and distance refer to the Hamming weight and distance. When we refer to the Lee weight and distance, it is always qualified. A code is nontrivial if |C| > 1. The minimum distance d of a nontrivial code C is min {d(x, y)|x, y ∈ C, x = y}. A code is e error correcting if its minimum distance d is at least 2e + 1. If Q is a group and C is a subgroup of Qn , then C is a group code. Group codes over nonabelian groups are asymptotically bad and not interesting [13]. However, there are many interesting abelian group codes. In particular, when Q is a finite field Fq and C is closed under vector addition, we have an additive code. If C is also closed under scalar multiplication, then it is a linear subspace of Fnq , and C is a linear code. Given two codes, it is natural to ask whether they are equivalent. For binary codes, it suffices to define two codes as isomorphic if one can be obtained from the other by a permutation of the coordinates only. Isomorphic codes are sometimes said to be permutation equivalent [11, p. 20] or permutationally isometric [1, p. 30]. In the general case, two codes are equivalent if, in addition to coordinate permutations, permutations of the symbol in the alphabet Q are also allowed. While equivalency preserves the distance distribution, it preserves neither linearity nor weights [6, p. 678]. Weights are preserved if the symbol 0 ∈ Q is fixed. Two codes are isometric if one can be obtained from the other by a combination of coordinate permutations, and permutations of the symbols in Q∗ = Q \ {0} [1, p. 29, p. 44]. Isometry still does not preserve linearity. Two linear codes over Fq are monomially equivalent or linearly isometric [1, p. 30] if one can be obtained from the other by a permutation of the coordinates, and by independent multiplications of the entries by a non-zero field element in Fq . Monomial equivalency preserves linearity. However, the linearity-preserving group may be larger. Any field automorphism of Fq , when applied simultaneous to the entries of all the codewords of a code also preserves linearity. When q = pr and r > 1, the Fronbenius automorphism of raising every element of Fq to its p-th power is not a monomial operation. In [1, Thm. 1.5.10], it was shown that monomially equivalency plus field automorphism give the full linearity preserving group, and it is called the group of semilinear isometries. Two codes are semilinearly isometric if one can be obtained from the other by using a semilinear isometry. Thus, for linear codes, we have three notions of equivalence: permutation, monomial(or linear), and semilinear. For binary linear codes, the three notions are the same. For linear codes over Fq where q is a prime, monomial and semilinear

280

C. Lam / Finding Error-correcting Codes Using Computers

equivalence are the same. When q is not a prime, but a prime power, the three are all different. The literature has many classification results on linear codes. Before one can compare these classification results, especially for the non-prime cases, one has to be careful which notion of equivalency is used. An automorphism of a code is an equivalence taking the code to itself. The set of automorphisms form a group. Of course, this group also depends on the equivalence being used. A code is characterized by three parameters, its length n, its size M = |C|, and its minimum distance d. Hence, it is denoted as an [n, M, d] code. For a linear code of dimension k over Fq , M = q k , and the code is denoted as an [n, k, d] code. Since M is normally large, and k is small, there is usually no confusion in using the two notations. When two of the parameters are fixed, it is also natural to ask for the best value of the third, giving rise to three optimization problems. Computers have been used extensively to determine these optimal values, to construct examples of codes attaining these values, and to enumerate them when possible. The objective of this paper is to survey where the boundary of current knowledge is and where further computer work may be helpful in expanding the status of knowledge about error-correcting codes. Since the area is vast, we need to be brief and selective. With the assumption that the boundary of knowledge is usually represented by the smallest unsolved cases, we shall try to list these cases, with the hope that further computational work may solve these cases. Before moving on, we should add that there is also an extensive body of computational tools and methodology developed which are applicable to problems in error-correcting codes. Interested readers should see [1,14].

Linear Codes One can count the number of semilinear isometry classes of linear codes of length n and dimension k over Fq without actually constructing them [1, Ch. 6]. However, there seems to be no easy way of counting the number of such classes with a given minimum distance d. Thus, the optimization problem of determining the maximum minimum distance when given the length n and dimension k is still difficult. Given n and k, we let Dq (n, k) denote the maximum minimum distance amongst all codes of length n and dimension k over Fq . Bounds for Dq (n, k) with small parameters can be found in [9]. Table 1 gives the smallest n for which Dq (n, k) is unsolved. Readers interested in this problem should consult [1, Ch. 9] and [14, Ch. 3, 4, and 6] for many relevant computational techniques.

Self-Dual Codes A special class of linear codes are the self-dual codes. Given x, y ∈ Fnq , the (Euclidean) inner product is

C. Lam / Finding Error-correcting Codes Using Computers q

n

k

Dq (n, k)

2 2

32 32

14 18

8-9 6-7

3

22

8

9-10

4

19

8

8-9

5

16

5

9-10

5

16

6

8-9

7

15

8

6-7

8

16

9

6-7

9

17

11

5-6

281

Table 1. Smallest unsolved minimum distance for linear codes

(x, y) =

n 

xi yi ,

n 

xi y¯i ,

i=1

and the Hermitian inner product is (x, y) =

i=1

where y¯i is the conjugate of yi . The dual of a code C is C ⊥ = {u ∈ Fnq |(u, v) = 0, ∀v ∈ C}. If C = C ⊥ then C is self-dual. A theorem of Gleason and Pierce [16, p. 200] implies that, in the following four cases, all the Hamming weights of a self-dual code over Fq are divisible by an integer c > 1: I II III IV

q q q q

= 2, c = 2, = 2, c = 4, = 3, c = 3, and = 4 with the Hermitian inner product, c = 2.

This theorem is so influential that the first three cases are often called Type I , Type II , and Type III codes. A Type I code is also called a singly-even code and a Type II code a doubly-even code. The two are not mutually exclusive, and a code is strictly Type I if it is not also of Type II. For codes over F4 , in addition to the Hermitian self-dual codes, Euclidean self-dual codes, and Additive self-dual codes have also been studied. The ordinary inner product is used for Euclidean self-dual codes; but the weight used is the Lee weight. If F4 = {0, 1, ω, ω 2 } where 1 + ω + ω 2 = 0, their respective Lee weights are {0, 1, 2, 1}. The Lee weight of a codeword in a Euclidean self-dual code is always even. It is a Type II code over F4 if the Lee weights of all its codewords are divisible by 4. It is a Type I code over F4 if some codewords have Lee weights not divisible by 4. The Gray map taking {0, 1, ω, ω 2 } to {00, 01, 11, 10}

282

C. Lam / Finding Error-correcting Codes Using Computers q

n

Highest-Bound

2(type I) 2(type II)

56 72

10 or 12 12 or 16

3

68

15-18

4(Hermitian) 4(Euclidian I)

32 24

10 or 12 8 or 10

4(Euclidian II) 4(Additive I)

24 14

8 or 12 5 or 6

4(Additive II)

24

8 or 10

5

20

8 or 9

7 20 9 or 10 Table 2. Smallest unsolved minimum distance for self-dual codes

maps Fn4 to F2n 2 . It takes Type I and Type II codes of length n over F4 to Type I and Type II binary codes of length 2n, respectively. As for Additive self-dual codes over F4 , the trace inner product is used: (x, y) =

n 

(xi yi2 + x2i yi ),

i=1

where x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). An Additive self-dual code is Type II if all its codewords are of even Hamming weight; otherwise, it is Type I. Technically, Additive self-dual codes are not linear codes, but they behave like a linear code, and they are important because of their relationship to quantum codes [5]. Tables of self-dual codes are maintained in [8,12,10]. Table 2 lists the smallest n for which the highest minimum distance is unsolved. Several upper bounds on the minimum distance of binary self-dual codes have been proved. The first one was given in 1973 [15]. Codes meeting these bounds are called extremal. The most famous open problem is probably the question whether an extremal doubly-even [72, 36, 16] code exists. There are many papers on the possible divisors of the order of its automorphism group. For a summary, see [12, p. 463]. Given a length n, codes with the maximum minimum distance have the best error correcting capability. Thus, we may want to classify all the self-dual codes of length n with the maximum minimum distance. Table 3 gives the smallest unsolved cases for which this classification is not complete. The column labelled “number” gives the number of inequivalent self-dual codes known. We may also want to classify all the self-dual codes for a given length n. Even partial results are useful. For example, the computer-aided proof of the non-existence of a (22, 8, 4)-BIBD was based on a complete classification of the self-dual binary codes of length 34 with minimum distance at least 4 [3]. The task of classifying self-dual codes is greatly facilitated by the mass formulae [16, p. 183-184]. A mass formula counts the number of self-dual codes for a given length and can be used to check the correctness of the classification. It can also be used to derive a lower bound for the number of equivalence classes by assumption all classes are of maximal size, which is equivalent to assuming

C. Lam / Finding Error-correcting Codes Using Computers q

n

minimum distance

number

2(type I) 2(type II)

38 40

6 8

≥ 900 ≥ 12579

3

32

9

≥ 239

4(Hermitian) 4(Additive I)

24 13

8 5

≥ 17 ≥9

4(Additive II)

14

6

≥ 491

5

18

7

≥1

283

7 16 7 ≥1 Table 3. Smallest unclassified self-dual codes with the largest minimum distance

q

n

lower bound(mass formula)

known

2(type I) 2(type II)

34 40

704 17493

≥ 20852 ≥ 12579

3

28

1001336

≥ 6931

4(Hermitian)

22

66265

≥ 723

4(Additive I) 4(Additive II)

13 14

72573550 1727942

≥ 1020

5

18

10930

7

16 261696 Table 4. Smallest unclassified self-dual codes

all codes are rigid. Thus, a lower bound for the number of inequivalent codes is obtained by dividing the number of self-dual codes by the size of the full transformation group. With the conjecture that when n is large, most codes are rigid, this lower bound also gives an estimate for the number of inequivalent codes. Table 4 gives the smallest n for which a complete classification of self-dual codes is unknown. As a guide to the size of the problem, the table lists a lower bound derived from the mass formula. It also lists the number of inequivalent code known so far. Tables of already classified self-dual codes can be found in [10,7]. Non-restricted Block Code Now, we consider the least restricted situation. Let |Q| = q, and let Aq (n, d) denote the maximum number of vectors from Qn with minimum (Hamming) distance d. Bounds for Aq (n, d) with small parameters can be found in [4]. Table 5 gives the smallest n for which there exists a gap between the lower and upper bounds for Aq (n, d). Conclusion This concludes a brief snapshot of where further computer work may be helpful in expanding the knowledge in this vast area of error-correction codes.

C. Lam / Finding Error-correcting Codes Using Computers

284

q

n

d

Aq (n, d)

2 2 2

17 17 17

4 5 6

2720-3276 512-680 256-340

2 2

17 17

7 8

64-72 36-37

3

7

3

99-111

4

6

3

164-179

5

7

3

1597-2291

5 5

7 7

4 5

250-545 53-108

Table 5. Smallest unsolved maximum size of unrestricted block codes

References [1]

[2] [3] [4] [5] [6] [7] [8] [9] [10]

[11] [12] [13] [14] [15] [16] [17]

A. Betten, M. Braun, H. Fripertinger, A. Kerber, A. Kohnert, and A. Wassermann, ErrorCorrecting Linear Codes - Classification by Isometry and Applications, Algorithms and Computation in Mathematics, 18, Springer, 2006. J. Bierbrauer, Introduction to Coding Theory, Chapman & Hall/CRC, 2005. R. T. Bilous and G. H. J. van Rees “Self-Dual Codes and the (22,8,4) Balanced Incomplete Block Design”, J. of Combin. Designs, 13(2002), 363-376. A. E. Brouwer, “Small table of bounds for binary/ternary/quaternary/5-ary codes.” Online available at http://www.win.tue.nl/~aeb/. Accessed on 2010-04-12. A. R. Calderbank, E. M. Rains, P. M. Shor, and N. J. A. Sloane, “Quantum error correction via codes over GF(4),” IEEE Trans. Inform. Theory IT-44 (1998), 1369-1387. C. J. Colbourn and J. H. Dinitz, eds., The CRC handbook of combinatorial designs, CRC Press Series on Discrete Mathematics and its Applications, CRC Press, 2006. E. Danielsen, “Database of Self-Dual Quantum Codes.” Online available at verb+http://www.ii.uib.no/ larsed/vncorbits/+. Accessed on 2010-04-14. P. Gaborit, “Tables of Self-dual Codes.” Online available at http://www.unilim.fr/pages_perso/philippe.gaborit/SD/. Accessed on 2010-04-14. M. Grassl, “Bounds on the minimum distance of linear codes and quantum codes.” Online available at http://www.codetables.de. Accessed on 2010-04-14. M. Harada and A, Munemasa, “Table of Self-dual Codes.” Online available at http://www.math.is.tohoku.ac.jp/~munemasa/selfdualcodes.htm. Accessed on 201004-13. W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge Univ. Press, 2003. W. C. .Huffman, “On the classification and enumeration of self-dual codes”, Finite Fields Appl., 11(2005), 451-490. J. C. Interlando, R. Palazzo, Jr., and M. Elia, Group Block Codes Over Nonabelian Groups are Asymptotically Bad, IEEE Trans. Info. Th, 42(1996), 1277–1280. ¨ P. Kaski and P. R. J. Ostergard Classfication Algorithms for Codes and Designs Algorithms and computations in mathematics, 15, Springer, (2006). C. L. Mallows, N. J. A. Sloane, “An upper bound for self-dual codes”, Inform. Control, 22(1973), 188-200. V. S. Pless and W. C. Huffman, eds., Handbook of Coding Theory, Elsevier, 2006. J. H. van Lint, Introduction to Coding Theory, 3rd Ed., Springer Grad. Texts in Math. 86, 1999.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-285

285

Quantum jump codes and related combinatorial designs Masakazu JIMBO a,1 and Keisuke SHIROMOTO b a Graduate School of Information Science, Nagoya University, Nagoya, Japan b Department of Mathematics and Engineering, Kumamoto University, Kumamoto, Japan Abstract. Quantum jump codes were introduced by Alber et al. (2001). Quantum jump codes have a close connection with combinatorial designs called t-SEED (tspontaneaus emission error design). In this paper, we give a brief survey of quantum jump codes together with some new results. Firstly, fundamental properties of a t-error correcting quantum jump code are described. Secondly, a few examples of jump codes are given and an upper bound on the dimension of a jump code with a fixed length and given error correcting ability is derived. A relation between a t-SEED and a jump code is discussed and various constructions of t-SEEDs are given. Keywords. quantum jump code, t-SEED, large set

Introduction Quantum error correcting codes have been studied by many authors [9,11,14,29,30] motivated by the pioneering work by Shor [28]. Among them, Alber et al. [1] introduced quantum jump codes which correct errors caused by quantum jumps. Quantum jump codes have a close connection with combinatorial designs called t-SEED (t-spontaneaus emission error design). In this paper, we give a brief survey of a quantum jump code together with some new results. In Sections 1 and 2, a breif introduction to a quantum jump code is given. In Section 3, a few examples of jump codes are shown and in Section 4, an upperbound of dimension of a jump code with a fixed length and given error correcting ability is explained. In Section 5, a non-existence result for a special parameter is shown. Moreover, in Section 6 a connection between a t-SEED and a jump code is discussed. Finally, in Section 7, various constructions of t-SEEDs are given.

1. Quantum codes We begin with the introduction of quantum error correcting codes. 1 Corresponding

Author: Masakazu Jimbo, E-mail: [email protected]

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M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

1.1. Quantum state A quantum state for a single particle can be represented by a vector in a finite dimensional Hilbert space H, that is, a vector space with inner product. In this paper we set H = C2 , where C is the set of complex numbers. A quantum state of a single quantum system like a photon is represented by |ϕ, called a ket vector, which is a 2-dimesional vector in H = C2 . The unit of the information amount for a single quantum system is called a ! ! 0 1 are called pure states and any state |ϕ is a and |1 = qubit. In particular, |0 = 1 0 linear combination (superposition) of these two pure states, that is, |ϕ = α|0+β|1 for α, β ∈ C. We define a bra vector ϕ| = |ϕ† , where |ϕ† is the tanspose of the complex conjugate of |ϕ. Then the inner product of |ϕ 9and |ψ is written by the notation ϕ|ψ and the size of a state vector |ϕ is written by ϕ|ϕ. In the field of quantum information, any state |ϕ and its scalar multiple α|ϕ (α = 0) are identified as the same quantum state. Hence, without loss of generality, we assume ϕ|ϕ = 1. A joint state of n-qubits is of the form |ϕ = |ϕ1 ϕ2 · · · ϕn  = |ϕ1  ⊗ |ϕ2  ⊗ · · · ⊗ |ϕn , where ⊗ is the tensor product. In this case, |ϕ is a 2n -dimensional vector in H⊗n = H ⊗ H ⊗ · · · ⊗ H. Let F = {0, 1} and F n = F × · · · × F. Then for any x = (x1 , x2 , . . . , xn ) ∈ F n , |x = |x1  ⊗ |x2  ⊗ · · · ⊗ |xn  are pure states for n-qubits and these 2n vectors in {|x : x ∈F n } form an orthonormal basis of H⊗n . Any n-qubit state can be represented by |ϕ = x∈F n αx |x.

1.2. State Transition

For any quantum state |ϕ ∈ H⊗n , a state transition can be represented by a linear operator. In a quantum computation or quantum data transmission, information is stored as a quantum state of an n-qubit system. Quantum computation can be pursued by applying suitable unitary operators. However, in these computation or data transmission system, we can not avoid the occurence of errors or noises caused by the interaction with environment. Because of the noise, the information stored in a quantum system may include some error. Errors or noises are also considered as operators. Typical unitary operators for a single qubit are σX

! 01 , = 10

! 1 0 , σZ = 0 −1

! 0 −i , σY = i 0

√ where i = −1. In order to correct such errors, we need to apply (inverse) unitary operators. But unlike to the classical data strage, we can not observe the quantum state of the system. Hence, we need to correct the quantum state by utilizing some partial information like an eigenvalue of a measurement, or by observing changes of outside of the system.

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

287

1.3. Quantum error correcting codes Let C be a subspace of H⊗n and E be a set of error operators including the identity operator. We assume that only the errors in E occur in a quantum system. C is called an E-error correcting quantum codes if, for any |c ∈ C and E ∈ E, we can recover the original state |c utilizing a partial information of E|c obatined by measurement without knowing the original state |c. For an E-error correcting quantum code C, the following theorem is known. Theorem 1 (Knill and Laflamme [23]) A subspace C ≤ H⊗n with orthonormal basis {|ci  : i = 1, . . . , m} is a quantum E-error correcting code if and only if the following holds: ci |E1† E2 |cj  = δij κE1 ,E2 ,

for any i, j, and E1 , E2 ∈ E,

(1)

where , 1, δij = 0,

if i = j, if i = j,

and κE1 ,E2 is a constant depeding only on E1 and E2 . Example 2 Let n = 2 and E = {I ⊗ I, E = I ⊗ σX }. Then, C = |00, |11 is an E-error correcting quantum code, since E|00 = |01, E|11 = |10 implies 00|E † E|11 = 0, 00|E|11 = 0, 00|E|00 = 11|E|11 = 0, 00|E † E|00 = 11|E † E|11 = 1, which satisfies the condition (1). The code space C is spanned by |00 and |11, whereas the space EC derived from error E is spanned by |01 and |10 as is seen in Figure 1. These subspaces are orthogonal and by measuring, that is, by checking eigenvalues of a set of projectors P0 = |0000| + |1111| and P1 = |0101| + |1010|, we find the subspace that the quantum state belongs. We can decode the received state by utilizing this information.

EC |01> |10>

|00> 0

C

|11> Figure 1. Code space and error space

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M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

2. A quantum jump code 2.1. A decay operator and a jump operator In this paper, we treat errors caused by spontaneous emission. Quantum state is changed according as the spontaneous emission by the loss of energy. In this case, there are two kinds of errors, that is, quantum decay and quantum jump. A quantum decay operator is represented by 2

− κt 2 |1 1|

D(t) = e

4 − κt 2

=e

3

0 05 01

! ! 00 10 − κt 2 , +e = 01 00

where t is a time variable and κ is a decay rate. Then, for x = 0, 1, D(t)|x =



! !

κt κt 00 10 |x = e−x 2 |x. + e− 2 01 00

(2)

holds. Assume that spontaneous decay occurs to each qubit with the same decay rate. Then the decay operator for n-qubit quantum state is defined by DV (t) = D(t)⊗· · ·⊗D(t) = D(t)⊗n . For any x = (x1 , x2 , . . . , xn ) ∈ F n , we have DV (t)|x =

n ; i=1

D(t)|xi  = e−wt(x) 2 |x, κt

(3)

where wt(x) is the Hamming weight of x, that is, the number of nonzero elements in x. On the other hand quantum jump is defined as follows: Let ! 01 , A = |01| = 00 then the quantum jump operator for a single qubit is defined by J|φ =

,

A|φ, |φ

if φ|A† A|φ = 0, if φ|A† A|φ = 0.

Thus, we have , β|0, J(α|0 + β|1) = |0,

if β = 0, if β = 0.

Remark: In Alber et al. [2], a jump operator is defined by 1 A|ϕ. J|ϕ = 9 ϕ|A† A|ϕ

(4)

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

289

Hence, in the case of |ϕ = |0, by setting |ϕ = ε0 |0 + ε1 |1, (|ε0 |2 + |ε1 |2 = 1), we should consider it as 1

J|0 = lim 9 ε1 −→0

ϕ|A† A|ϕ

A|ϕ = |0.

9 In this paper, we ignore the normalizing denominator ϕ|A† A|ϕ and instead we defined that any state vector |ϕ is identified with its scalar multiple in Subsection 1.1. In the case of n-qubit system, a jump operator at the i-th position is defined by Ji = I ⊗ I ⊗ · · · ⊗ I ⊗ J ⊗ I · · · ⊗ I. Let V = {1, 2, . . . , n}. If jump error operators Ji1 , . . . , Jis−1 , Jis are applied in turn to a quantum state |c, such multiple jump is represented by JE = Jis Jis−1 · · · Ji1 , where E = (i1 , · · · , is ) is an ordered s-tuple (s-list). In general, jump operators Ji1 , . . . , Jis−1 , Jis are not commutative. For example, J2 J1 (|101 + |010) = |001, whereas J1 J2 (|101 + |010) = |000. However, for a state |c, by deleting jump operator Jij ’s which do not change the state Jij−1 · · · Ji1 |c we can get a subsequence of operators JEc = Jijr ·  · · Jij1 , where Ec = (ij1 , · · · , ijr ) ⊂ E. Hence JE |c = JEc |c holds for the state |c = x∈F n αx |x and there are x’s such that αx = 0 and supp(x) ⊃ Ec hold, where supp(x) = {i : xi = 0} for x = (x1 , . . . , xn ). Moreover, the operators in JEc are commutative when it is applied to |c. Hence, for a multiple jump operator JE and a state |c, we have only to consider multiple jump operators which are commutative with respect to |c. Now, for a subset E of V , when the jumps at positions in E are commutative for |c, we denote it by JE =

n ; i=1

Ai ,

Ai =

,

I J

if i ∈ / E, if i ∈ E.

The position where a quantum jump occured can be detected by the continuous monitoring of photodetector since a photon is radiated when a quantum jump occured at a qubit (see Figure 2). Hence, we assume that the positions where quantum jumps occur are known (see, Alber et al. [2]). In general, a decay and jump process is written as DV (ts ) · Jis · DV (ts−1 ) · Jis−1 · · · · · DV (t1 ) · Ji1 · DV (t0 ). That is, as it is shown in Figure 3, within a time period t, spontaneous decay occurs to each qubit with the same decay rate and among the period quantum jumps occur s times. 2.2. Decoherence-free subspace for decay operator Our aim in this paper is to construct a code which can correct errors caused by quantum decay and quantum jumps. For quantum decay error, we apply a passive error correction, that is, we consider the error-free space caused by spontaneous decay error operator DV (t).

290

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs a quantum jump

a photon

n qubits

photodetector

Figure 2. A quantum jump and photodetector

t0

t1

ts-1

ts

0

 

t jump at i1-th jump at i2-th position position

jump at is-1-th position

jump at is-th position

Figure 3. Decay and jump process

Hence, we find a subspace W in which every quantum state vector is invariant with respect to the state transition by DV (t). A subspace W is called a decoherence-free subspace if DV (t)|ϕ = α|ϕ holds for any |ϕ ∈ W , where α is a nonzero constant. Now, let Fkn = {x ∈ F n : wt(x) = k} and let Wk =< |x : x ∈ Fkn > be a subspace which is spanned by {|x : x ∈ Fkn }. Lemma 3 W is a decoherence-free subspace with respect to a decay operator DV (t) if and only if W is a subspace of Wk for an arbitrary fixed weight k.  Proof. For any |ϕ = x∈F n αx |x ∈ W ,  (i) αx DV (t)|x DV (t)|ϕ = x∈F n  (i) κt αx e−wt(x) 2 |x = x∈F n =

n  

k=0 x∈Fkn

(i)

αx e−

κkt 2

|x

holds. In order that DV (t)|ϕ = const.|ϕ holds for any t, weight k must be constant, which prove the lemma.  Hence, any quantum jump code C must be in a subspace of Wk for some k to ignore the quantum decay error. Furthermore, for a quantum state |c ∈ Wk and a jump operator JE , JE |c ∈ Wk′ holds for some k ′ ≤ k.

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291

2.3. Quantum jump codes If we want to find an “e-error correcting” quantum jump code as a subspace of Wk , then we have only to consider error operators of the form E = {JE : E is a s-list of elements in V , s ≤ e}. In quantum jump codes, it is assumed that the positions E = (i1 , i2 , . . . , is ) of quantum jump occured are known by the continuous observation of photodetector as it was stated in Section 2.1. Note that if the error positions are known, the conditions (i) and (ii) in Theorem 1 are simplified as ci |JE† JE |cj  = δij κE

for any i = j and JE ∈ E,

(5)

where κE is a nonzero constant depending only on E. A subspace of Wk satisfying (5) is called an e-error correcting quantum jump code, denoted by an (n, m, e)k jump code, where m is the dimension of C. For a vector x = (x1 , x2 , . . . , xn ) ∈ F n and E = {ℓ1 , ℓ2 , . . . , ℓs }, let x|E = (xℓ1 , xℓ2 , . . . , xℓs ). Lemma 4 Let C be an (n, m, e)k jump code. Then for any E, with |E| = s ≤ e, and for any y ∈ F s , the following hold: (i)

(i) JE |ci  = |ci  implies that αx = 0 for any x ∈ Fkn such that x|E = (1, 1, . . . , 1). (ii) For an orthonormal basis {|ci  : i = 1, . . . m}, if JE |ci  = |ci  holds for some i then JE |cj  = |cj  holds for any j = 1, . . . , m. Proof. (i) is obvious. (ii) holds since JE |ci  =  |ci  implies that ci |JE† JE |ci  < ci |ci  = 1.  3. Examples of 1- and 2-error correcting quantum jump codes 3.1. A 1-error correcting quantum jump code of length four Here, we consider an example of 1-error correcting quantum jump codes of length four. Let C be a 1-error correcting quantum jump code. A codeword |c is represented by |c =



x∈Fkn

αx |x

for some fixed weight k, 0 ≤ k ≤ 4. Now, let  (i) αx |x : i = 1, 2, . . . , m} {ci = x∈Fkn be an orthonormal basis of C. Since, ci |cj  = δij ,

292

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs



x

(i) (j)

(6)

αx αx = δij

∈Fk4

(i)

holds for each i and j, and for a fixed k ∈ {0, 1, . . . , 4}, where αx is the complex (i) conjugate of αx . Similarly,

Jℓ |ci  =





(i)

x∈Fk4

αx Jℓ |x =

x∈Fk4 ,xℓ =1

(i)

αx |Pℓ x,

where Pℓ is the 4 × 4 diagonal matrix whose diagonal elements are 1 except for the ℓ-th element being 0. Hence,

ci |Jℓ† Jℓ |cj  =



(i) (j)

αx αx = δij κk,ℓ,1

(7)

x∈Fk4 ,xℓ =1

holds for each i, j and ℓ ∈ V , where κk,ℓ,1 is a nonzero constant depending only on k and ℓ. Hence, (6) and (7) can be rewrited as 

(i) (j)

(8)

(i) (j)

(9)

αx αx = δij κk,ℓ,1

∈Fk4 ,xℓ =1

x



αx αx = δij κk,ℓ,0

x∈Fk4 ,xℓ =0 for any i, j and ℓ. Hence, we can easily see that the weight k of the decoherence-free subspace Wk must be 2, since

(i)

α0001 = α0010 = α0100 = α1000 = 0,

(i)

α1110 = α1101 = α1011 = α0111 = 0

α0000 = 0, α1111 = 0,

(i)

(i)

(i)

(i)

(i)

(i)

(i)

(i)

hold by (8) and (9). Moreover we have the following equations for any i and j by (8) and (9):

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs (i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

293

α1100 α1100 + α1010 α1010 + α1001 α1001 = δij κk,1,1 , α1100 α1100 + α0110 α0110 + α0101 α0101 = δij κk,2,1 , α1010 α1010 + α0110 α0110 + α0011 α0011 = δij κk,3,1 , α1001 α1001 + α0101 α0101 + α0011 α0011 = δij κk,4,1 , α0011 α0011 + α0101 α0101 + α0110 α0110 = δij κk,1,0 , α0011 α0011 + α1001 α1001 + α1010 α1010 = δij κk,2,0 , α0101 α0101 + α1001 α1001 + α1100 α1100 = δij κk,3,0 , α0110 α0110 + α1010 α1010 + α1100 α1100 = δij κk,4,0 . By solving these equations for i = j, we can find the following relations: (i)

(i)

(i)

|α1100 | = |α0011 | = w1 , (i) 2

w1

(i) 2

+ w2

(i) 2

+ w3

(i)

(i)

(i)

|α1010 | = |α0101 | = w2 ,

(i)

(i)

(i)

|α1001 | = |α0110 | = w3 , (10)

= const.

Moreover, in the case of i = j, we have (i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

α1100 α1100 = α0011 α0011 ,

(i)

(j)

(i)

(j)

α1010 α1010 = α0101 α0101 , (i)

(i)

(j)

(i)

(j)

α1100 α1100 + α1010 α1010 + α1001 α1001 = 0. A solution satisfying (10), (11) can be obtained as follows: |ci  = ui1 |h1  + ui2 |h2  + ui3 |h3  for i = 1, 2, 3, where 1 |h1  = √ (|1100 + eiθ1 |0011), 2 1 |h3  = √ (|1001 + eiθ3 |0110) 2

1 |h2  = √ (|1010 + eiθ2 |0101), 2

and ⎛

⎞ u11 u12 u13 U = ⎝u21 u22 u23 ⎠ u31 u32 u33

is a unitary matrix. In particular, let θi = 0 for any i and let U = I, then 1 |c1  = √ (|1100 + |0011), 2 1 |c3  = √ (|1001 + |0110) 2

(j)

α1001 α1001 = α0110 α0110 ,

1 |c2  = √ (|1010 + |0101), 2

(11)

294

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

is an example of a 1-error correcting quantum jump code of length 4. It is easy to show that the code C with an orthonormal basis (ONB) {|c1 , |c2 , |c3 } has the maximum dimension. In fact, as we saw that any 1-error correcting jump code C of length 4 is a subspace of a space spanned by {|x|x ∈ F24 }. Moreover, after a jump error Jℓ occured, Jℓ C still has to have the same dimension with C because Jℓ |ci  must be  (1) orthogonal for any i. Let ℓ be a position where a codeword |c1  = x∈F 4 αx |x has a 2

(1)

term that αx = 0 and x has 1 in the position ℓ. In this case, c1 |Jℓ† Jℓ |c1  < c1 |c1  = 1.  (i) Since ci |Jℓ† Jℓ |ci  = c1 |Jℓ† Jℓ |c1  holds for any |ci  = x∈F 4 αx |x, there must be a 2 (i) vector |x such that |x has 1 at position ℓ and αx = 0. Thus, Jℓ C is spanned by the ket vector whose weight is one and xℓ = 0. There are three such vectors of weight 1 whose ℓ-th element is 0. Thus dim C ≤ 3 holds.

3.2. An example of 2-error correcting quantum jump codes of length 6 Here, we consider an example of 2-error correcting quantum jump codes of length 6. Let C be a 2-error correcting quantum jump code. And let {ci =



(i)

x∈Fk6

αx |x : i = 1, 2, . . . , m}

be an orthonormal basis of C. Let E = {ℓ1 , ℓ2 } be the set of positions where jump errors occur. If there are some (i) x such that supp(x) ⊃ E and αx = 0, we have JE |ci  =



x∈Fk6

=

(i)

αx JE |x 

x∈Fk6 ,xℓ =1for ℓ∈E

(i)

αx |PE x,

where PE is the 6 × 6 diagonal matrix whose diagonal elements are 1 except for the positions in E being 0. Hence, ci |JE† JE |cj  =

x



(i) (j)

αx αx = δi,j κE ,

(12)

∈Fk6 ,xℓ =1for ℓ∈E

where κE is a nonzero constant depending only on E. (i) By (12), it is shown that αx = 0 for any x with wt(x) = 0, 1, 2, 4, 5, 6. Thus, in this case a decoherence-free subspace is W3 . The following is an example of (6, 2, 2)3 jump code. Example 5 A (6, 2, 2)3 jump code is given by the following orthonormal basis:

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

295

1 |c1  = √ (|111000 + |101100 + |100110 + |100011 + |110001 10 + |011010 + |001101 + |010110 + |001011 + |010101),

1 |c2  = √ (|000111 + |010011 + |011001 + |011100 + |001110 10

+ |100101 + |110010 + |101001 + |110100 + |101010). It can be checked that (12) holds for any E ⊂ V , |E| ≤ 2. For example, let E = {1, 2}, then 1 JE |c1  = √ (|001000 + |000001) and 10 1 JE |c2  = √ (|000010 + |000100) 10 hold, which imply that ci |JE† JE |ci  = 51 for i = 1, 2 and c1 |JE† JE |c2  = 0. Similarly, for any E with two elements, JE |ci  consists of two basis ket vectors. Also, for any E with a single element, it consists of five basis vectors. These facts implies that ⎧1 ⎪ 5, ⎪ ⎪ ⎨1, ci |JE† JE |cj  = 2 ⎪ 1, ⎪ ⎪ ⎩ 0,

if i = j and |E| = 2, if i = j and |E| = 1, if i = j and E = φ, if i = j.

Remark: As you will see later, |c1  and |c2  are derived from two disjoint “2-(6, 3, 2) designs”, which include all triples from V .

4. An upper bound for the dimension of jump codes In this section, fundamental properties of an (n, m, e)k jump code are described. Most of the results in this section, we refer the reader to Beth et al. [8]. For a vector x = (x1 , x2 , . . . , xn ) ∈ F n and E = {ℓ1 , ℓ2 , . . . , ℓs }, let x|E = (xℓ1 , xℓ2 , . . . , xℓs ). Lemma 6 Let C be an (n, m, t)k jump code. Then for any E, with |E| = s ≤ t, and for any y ∈ F s , 

∈Fkn ,

x

(i) (j)

αx αx = δij κE,y

(13)

x|E =y

holds, where κE,y is a constant depending only on E and y. Proof. In the case of y = 1s = (1, 1, . . . , 1) ∈ F s for s ≤ t, (13) is ovbious by (5). Note that, even in the case of JE |ci  = |ci , (13) holds with κE,1s = 0 by (i).

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

296

We prove (iii) by induction for the weight of y. For E ⊂ V such that |E| = s < t and a vector y ∈ F s with weight w < s, without loss of generality, we assume that the first w elements are 1 and the other s − w elements are 0. Let E0 = {i : yi = 1} and we define  (i) (j) αx αx . N (E, y) = x∈Fkn x|E =y Then, N ((E, y) = N (E0 , 1w ) −



N (E, (1w , z))

z ∈F s−w ,z =0

holds, where (1w , z) is a concatination vector of 1w and z. In the case when i = j, each term in the right hand side of the above equation is constant by the induction assumption. Similarly, when i = j, each term in the right hand side is 0, hence the lemma is proved.  The following lemma is a direct consequence of Lemma 6. ⊗n Lemma 7 (Beth et al. [8]) If C is an (n, m, t)k jump code, then σX C is an (n, m, t)n−k jump code.

Lemma 8 (Beth et al. [8]) If an (n, m, t)k jump code exists for k > t > 1, then an (n − 1, m, t − 1)k−1 exists. Proof. The lemma can be obtained by applying an error operator Jn to the (n, m, t)k jump code C. Note that if {|ci  : i = 1, . . . , m} is an othonormal basis then {Jn |ci  : i = 1, . . . , m} is also an othonormal basis.  Lemma 9 (Beth et al. [8]) If an (n, m, t)k jump code exists for k > t ≥ 1, then an (n + 1, m, t)k jump code and an (n + 1, m, t)k+1 jump code exist. Proof. Appending |0 or |1 to a an (n, m, t)k jump code, an (n + 1, m, t)k or an (n + 1, m, t)k+1 jump code can be obtained, respectively. The following upperbound is obtained by Beth et al. [8]. Proposition 10 (Beth et al. [8]) The dimension m of a (n, m, t)k jump code is bounded by m ≤ min

-



.

n−t n−t n−t . ≤ , ⌊n/2⌋ − t k k−t

(14)

  Proof. It is ovbious that (n, m, 0)k jump code has dimension dim Wk = nk . If C is an (n, m, t)k jump code, then by Lemma   8, a JE C is an (n − t, m, 0)k−t jump code for E ⊂ V , |E| = t. Hence, dim C ≤ n−t k−t . Also, by Lemma 7, an (n, m, t)n−k jump code

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

exists, hence an (n − t, m, 0)n−k−t jump code exists, which means dim C ≤ n−t k .



n−t n−k−t

297



= 

the upperbound (14) exists Lemma 11 (Beth et al. [8]) An (n, m, 1)k jump code attaing   for any even integer n. In the case, k = n2 and m = 21 nk .

Proof. Let |cx  = √12 (|x + |x) for any x ∈ F n , wt(x) = n2 , where x = (1 − x1 , 1 −x2, . . . , 1 − xn ). Then, the code C with orthonormal basis {|cx } has dimension m = 21 nn . And 2

Ji |cx  =

,

√1 |x, 2 √1 |x, 2

if i ∈ supp(x) , if i ∈ / supp(x)

holds. Hence cx |Ji† Ji |cx′  = 21 δx,x′ , which prove the lemma. 5. Non-existence of a (6, 3, 2)3 jump code By the upperbound (14), dim C = m ≤ 4 holds for a (6, m, 2)3 jump code C. Moreover, Beth et al. [8] showed that there does not exist a (6, 4, 2)3 jump code. Here we show that there does not exist a (6, 3, 2)3 jump code. Lemma 12 There is no (6, 3, 2)3 jump code. Proof. Assume that there are three orthonormal vectors |c1 , |c2 , |c3  which span the basis of a (6, 3, 2)3 jump code. Case 1: Firstly, we consider the case when there is a coordinate ℓ such that Jℓ |ci  = |0 holds for every i = 1, 2, 3. In this case, by deleting the ℓ-th coordinate, C can be viewed as a subspace of H⊗5 . That is, we consider the existence of a (5, 3, 2)3 jump code. For W35 =< |x : x ∈ F35 >, let |ci  =



x

∈W35

(i)

αx |x

for i = 1, 2, 3. Then, by setting E = {1, 2} and y = (00) in Lemma 6, we obtain (1)

(2)

α00111 α00111 = 0,

(2)

(3)

α00111 α00111 = 0,

(3)

(1)

α00111 α00111 = 0

and (1)

(1)

(2)

(2)

(3)

(3)

α00111 α00111 = α00111 α00111 = α00111 α00111 , (i)

(i)

which implies α00111 = 0 for any i. By a similar argument, we get αx = 0 for any x ∈ F35 and i = 1, 2, 3. Hence there does not exist a (5, 3, 2)3 jump code. Case 2: Secondly, we consider the case when there are no coordinate ℓ such that Jℓ |ci  = |0 for every i = 1, 2, 3. In this case, vector ci ’s are linear combinations of twenty vectors in W3 = {|x : x ∈ F36 }. Let

298

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

|ci  =



x∈W3

(i)

αx |x

for i = 1, 2, 3. Without loss of generality, we choose vectors |111000 and |000111. Then similarly to (8) and (9), we obtain the following equations including the term of (i) (i) α111000 and α000111 : (i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

(i)

(j)

α111000 α111000 + α110100 α110100 + α110010 α110010 + α110001 α110001 = 0 α111000 α111000 + α101100 α101100 + α101010 α101010 + α101001 α101001 = 0 α111000 α111000 + α011100 α011100 + α011010 α011010 + α011001 α011001 = 0 α000111 α000111 + α001011 α001011 + α001101 α001101 + α001110 α001110 = 0 α000111 α000111 + α010011 α010011 + α010101 α010101 + α010110 α010110 = 0 α000111 α000111 + α100011 α100011 + α100101 α100101 + α100110 α100110 = 0. Summing up all these equations and by subtracting 

(i) (j)

αx αx = 0,

x∈W3 we obtain (i)

(j)

(j)

(i)

α111000 α111000 = −α000111 α000111 . for any i = j. This can be shown for any x ∈ W . Hence, (i) (j)

(i)

(j)

(15)

αE αE = −αE c αE c

    for any E ∈ V3 and i = j, where E c = V \ E and Vk is the set of k-element subsets of V . By applying the similar calculation to (8) and (9), we obtain (i)

(i)

(j)

(j)

|αE |2 + |αE c |2 = |αE |2 + |αE c |2

(16)

  for any E ∈ V3 and i = j. By multiplying (15) for (i, j) = (1, 2), (2, 3), (3, 1), we have (1) (2) (3)

(1) (2) (3)

|αE αE αE |2 = −|αE c αE c αE c |2 , which means (1) (2) (3)

(1) (2) (3)

αE αE αE = αE c αE c αE c = 0. (1)

(1)

(i)

(i)

Case 1. The case of αE = 0 and αE c = 0: In this case, by (16) αE = 0 and αE c = 0   for any E ∈ V3 and i = 1, 2, 3. Hence, |ci  = 0 for any i, contradiction.

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs (1)

(1) (3)

(2)

299

(2) (3)

Case 2. The case of αE = 0 and αE c = 0: In this case, αE c αE c = αE αE = 0 holds (3) (2) (3) (1) by (15). The case of αE c = 0 or αE = 0 results in Case 1. Hence, αE c = αE = 0, which also results in Case 1. Hence, the lemma is proved.  By Lemma (12), we found that a (6, 2, 2)3 jump code in Example 5 has the maximum possible dimension for n = 6, k = 3 and t = 2.

6. A t-SEED and a jump code  (i) (i) Though the coefficients αx of a ket vector |c = x∈F n αx |x are complex numbers (i) in general, by restricting the values of αx to 0 and α, where α is a normalizing constant satisfying c|c = 1, the combinatorial structure of quantum jump codes are closely related to combinatorial designs. Here, we identify a vector x = (x1 , x2 , . . . , xn ) ∈ F n with its support set B = supp(x) = {i : xi = 1} and |x with |B. Then a ket vector |c = x∈F n αx |x is represented by 1  |B, |c = |B = 9 |B| B∈B

where B = {supp(x) : αx = 0}. Now, let V /E be the family of subsets of V including E ⊂ V . We define projection matrices Mk = LE =



|xx| =

x∈Fkn 

E⊂supp(x)



B∈(

V k

|xx| =

)

|BB| and 

B∈V /E

for any 0 ≤ k ≤ n and E ⊂ V , then for any state |c = Mk |c = LE |c =

|BB|

 x∈F n αx |x,

 1 αx |x = 9 |B, |B| x∈Fkn B∈B∩(Vk ) 

 1 |B αx |x = 9 |B| B∈B∩(V /E) supp(x)⊃E 

hold. By using Mk and LE , the conditon (5) for a t-error correcting quantum jump code with orthonormal basis {|ci } can be characterized by (i) ci |Mk |ci  = 1 for any i and for given k (t < k < n − t), (ii) ci |LE |cj  = δij λE for any i, j and E ⊂ V such that |E| ≤ t. For an orthonormal basis {|ci  = |Bi  : i = 1, 2, . . . , m}, by noting

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

300



1

ci |LE |cj  = 9

|B(i) | · |B(j) | B∈B(i) ∩B(j)∩(V /E)

we find that (i) and (ii) implies

1=

|B(i) ∩ B(j) ∩ (V /E)| 9 , |B(i) | · |B(j) |

(T1) |B| = k for any B ∈ B(i) , |{B∈B(i) :B⊃E}| |B(i) |

= λE holds for any i and E ⊂ V , |E| ≤ t, where λE is a constant depending on E. (T3) B (i) ∩ B (j) = φ for i = j.   For an n-set V and B (i) ⊂ Vk , (i = 1, . . . , m), if (T1), (T2), (T3) are satisfied, then a system (V ; B(1) , . . . B (m) ) is called a t-spontaneous emission error design, denoted by t-(n, k; m) SEED (see Figure 4). (T2)

V   k   

B(1) V E

1 0 0 1 1 1 0 1

1 0 0

D(1)

B(2)

B(3)

B (m)

D(2)

D(3)

D(m) not used

0 1 1 1 0 0 1

λE|B(1)|

λE|B(2)|

λE|B(3)|

λE|B(m)|

Figure 4. An incidence matrix of a t-SEED

Note that when λE depends only on the number of elements in |E|, a pair (V, B (i) ) is called a t-(n, k, λ) design, where λ = λE for |E| = t. In particular, a t-(n, k, 1) design t-design, denoted by S(t, k, v). Moreover if |B| is constant  *mis called aSteiner and i=1 B (i) = Vk , a t-SEED is called a large set of a t-(n, k, λ) design, denoted by v−t LSλ (t, k, n). The number of t-designs in a large set is m = k−t /λ. Lemma 13 For a fixed k ≤ tion 10.

n 2,

an LS1 (t, k, n) attains the upper bound (14) of Proposi-

7. Constructions of t-SEEDs In this section various constructions of t-SEEDs are described. 7.1. Large sets Firstly, known large sets are listed here. For deatils of large sets, we refer the reader to Khosrovshahi and Tayfeh-Rezaie [20], Colbourn and Dinitz [13] and Tierlinck [31].

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

301

(i) (ii) (iii) (iv) (v) (vi) (vii)

An LS1 (2, 3, v) exists for all admissible parameters of v = 7. An LSλ (3, 4, v) exists for v ≡ 0 (mod 3). min (4, 5, 20v + 4) exists for gcd(v, 30) = 1. An LSλ min An LS60 (4, 5, 60v + 4) exists for gcd(v, 60) = 1, 2. The number of disjoint designs in LSλ (t, t + 1, v) is ℓ = v−t λ . No LS1 (3, 4, v) is known. Etizon and Hartman(1991) obtained near large set with v − 5 disjoint 3-(v, 4, 1) designs for v = 5 · 2n . (viii) An LS3 (3, 4, v) exists for v ≡ 0, 6 (mod 12). (ix) An LS6 (3, 4, v) exists for v ≡ 9 (mod 12). (x) An LS12 (3, 4, v) exists for v ≡ 3 (mod 12). 7.2. t-SEEDs derived from orthogonal arrays Let S be a set of q elements. A q t × k array A with elements in S is called an orthogonal array, denoted by OA(t, k, q), if each ordered t-tuple occurs exactly once in any t-columns of A. A large set of an orthogonal array LOA(t, k, q) is a collection {Ar }r∈R of OA(t, k, q)’s such that every ordered k-tuple of S occurs exactly once in one of Ar . Note that |R| = q k−t . The following is known (see Raghavarao [26]): Proposition 14 If there is an OA(t, k, q), then there is a large set LOA(t, k, q). By this Proposition, we obtain the following: Theorem 15 If there exists an OA(t, k, q), then there exists a t-(kq, k; q k−t ) SEED. Example 16 If q is a prime power, then there exists a t-(qk, k; q k−t ) SEED for k ≤ q+1. Remark: Beth et al. [8] obtained a t-SEED for k = q. Moreover, Beth et al. [8] claimed that log(dim. of jump code by Theorem 1) (q − t) log q =  2 −t −→ 1 log(the upper bound of (14)) log qq−t

as q −→ ∞ for fixed t. On the other hand, it holds that

dim. of jump code by Theorem 1 q q−t = q2 −t −→ 0 the upper bound of (14)

(17)

q−t

as q −→ ∞ for fixed t. Hence, we may pose a question whether there is a sequence of t-SEEDs which is asymptotically optimal in the sence that (17) tend to 1 except for a series of large sets? 7.3. Product methods and recursive constructions (ℓ)

Let (V1 ; B (1) , . . . , B (m) ) be a t-(n, k; m) SEED and k × q t matrices A(ℓ) = (aij ) be an LOA(t, k, q) with elements {0, 1, . . . , q − 1}. Let V = V1 × {0, 1, . . . , q − 1} and construct families of blocks

302

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs (ℓ)

(ℓ)

B(h,ℓ) = {{(b1 , a1j ), . . . , (bk , akj )} : (b1 , . . . , bk ) ∈ B(h) , j = 1, . . . , q t } for h = 1, . . . , m, ℓ = 1, . . . , q k−t . Then, we obtain the following theorem. Theorem 17 If there are a t-(n, k; m) SEED and a LOA(t, k, q), then there is a t(nq, k; mq k−t ) SEED. Applying this recusive construction to Theorem 15, we obtain the following: Corollary 18 For a prime power q, a t-(q n (q + 1), q + 1; q n(q+1−t) ) SEED exists. Beth et al. [8] gave a construction which combines a quantum jump code and a usual quantum code. Theorem 19 (Beth et al. [8]) Let C = (n, p, t)k be a jump code of prime dimension. Furthermore, let Cp = [[N, K, D]]p be a “quantum error-correcting code” in the space (Cp )⊗N . Then the concatenation of C as inner and Cp as outer code yields a jump code C = (N n, pK, T )N w on N n-qubits with T ≥ D(t + 1) − 1. A t-(n, k; m) SEED (V ; B (1) , . . . , B (m) ) is said to be s-resolvable if each B (i) is partitioned into h subfamilies B(i,1) , . . . , B (i,h) and a (V ; B(1,1) , B(1,2) , . . . , B (m,h) ) forms an s-(n, k; mh) SEED. Theorem 20 If there is a ⌊ 2t ⌋-resolvable t-(n, k; m) SEED (V ; B (1) , . . . , B(m) ), then there exists a t-(nv, 2k; hm2 ) SEED for any v ≥ 2, where h is the number of subfamilies B (i,j) in B(i) . We will give an example of Theorem 20. Let Kn be the complete graph of order n. For even n, a 1-factor of Kn is a set of independent edges. A 1-factorization of Kn is a partition of the edges of Kn into n − 1 one-factors. For any even n, there exists a 1-factorization of Kn . A 1-factorization can be seen as a 1-resolvable 3-(n, 2; 1) SEED. Hence, by Theorem 20, we obtain the following corollary. Corollary 21 For any even n and for any integer v ≥ 2, there is a 3-(nv, 4; n−1) SEED, which is an (nv, n − 1, 3)4 jump code. Remark: In this case, the upper bound of the dimension is nv − 3. When v ≤ 3 this is better than that of Corollary 2 for k = 4, t = 3. Example 22 In Figure 5 and Table 1, a 1-factor for K4 is presented. A column of Table 1 corresponds to an edge. And any two columns partitioned by vertical lines correspond to 1-factors. In Table 2, each column corresponds to a block. Let V = {00 , 10 , 20 , 30 , 01 , 11 , 21 , 31 }, then a four tuple (a, b|c, d) means a block {a0 , b0 , c1 , d1 }.

M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs 0

1

2

0

2

0

2

0

3

1

3

1

3

1

Factor 1

K4

303

2

3 Factor 3

Factor 2

Figure 5. A 1-factor for n = 4

Table 1. A 1-factor for n = 4 0 1

2 3

0 2

1 3

0 3

1 2

Table 2. A 3-(8, 4; 3) SEED for n = 4, v = 2, m = 1, h = n − 1 = 3 002200110011 113322333322

002200110011 113322333322

002200110011 113322333322

020201010101

010101010202 232332321313

010102020101 323213132323

131323233232

7.4. 2- and 3-SEEDs derived from affine geometry It is well known that the set of planes in AG(n, q) yields a 2-(v = q n , k = q 2 , λ = (q n−1 − 1)/(q − 1)) design. Lemma 23 The 2-design generated by the set of 2-flats in AG(n, q) is decomposed into (i) (ii)

q n−1 −1 q 2 −1 q n−1 −q q 2 −1 n

number of 2-(v = q n , k = q 2 , λ = q + 1) designs when n is odd.

number of 2-(v = q n , k = q 2 , λ = q + 1) designs and one 2-(v = q , k = q 2 , λ = 1) design when n is even.

Munemasa [25] showed better results for q = 2 by examining the orbit structure of PG(2n − 1, 2). Lemma 24 (Munemasa [25]) The number of lines in PG(2n − 1, 2) whose orbit under the subgroup of index 3 in the Singer group is a spead is given by 1 2n (2 − 1)(2n + (−1)n+1 )2 . 27 By using Lemma 24, we can obtain a 2-SEED. Example 25 PG(7, 2) has 28 − 1 = 255 points and

(18)

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(28 − 1)(27 − 1) = 255 × 42 + 85 (22 − 1)(2 − 1) lines. These lines are partitioned into 42 Singer cycles whose orbits are full and a single cycle of short orbit with orbit length 85. For a line L in a full orbit, {a(L ∪ 0) + b : a ∈ GF (28 )× , b ∈ GF (28 )} generates a 2-(28 , 4, 3) design and a line in the short orbit generates a 2-(28 , 4, 1) design. Among those 42 full orbits there are 8 orbits each of which can be partitioned into 3 spreads of lines. Actually, for a root β of the primitive irreducible polynomial x8 + x5 + x3 + x2 + x + 1, B = {β 0 , β 7 , β 173 } and its Frobenius cycle of length 8 are lines in such orbits. Hence, we obtain 24 spreads and each of these spreads generates a 2-(28 , 4, 1) design. As a total, we obtain (24 + 1) 2-(28 , 4, 1) designs and (42 − 8) 2-(28 , 4, 3) designs, which generate a 2-(28 , 4; 59) SEED. In general, by Lemmas 23 and 24 it holds that the number of full orbits are Among these, there are

22n−1 −2 . 3

(2n + (−1)n+1 )2 − 9 27 orbits which can be partitioned into 3 spreads. Hence, we obtain the following theorem. Theorem 26 The 2-design generated by the set of 2-flats in AG(2n, 2) is decomposed into 22n−1 − 2 (2n + (−1)n+1 )2 − 9 − 3 27 number of disjoint 2-(22n , 4, 3) designs and (2n + (−1)n+1 )2 −1 9 number of disjoint 2-(22n , 4, 1) designs. Hence, there is a 2-(22n , 4; f2n ) SEED, where f2n =

22n−1 − 2 2{(2n + (−1)n+1 )2 − 9} +1+ . 3 27

Now, for V = GF(2)n , let σ be a mapping such taht σ : x −→ xs for x ∈ V . Then our problem is to find the condition on s in order that a D and σ(D) are disjoint, where i D is a 3-design generated from 2-flats of AG(n, 2). When s = 2i , σ : x −→ x2 is a Frobenius automorphism of D. In this case, it holds that σ(D) = D. Let D = (V, B) is the 3-(2f , 4, 1) design derived from 2-flats of AG(f, 2). Let s be an integer such that (s, n) = 1, where n = 2f − 1. Then σ : x −→ xs is a bijection on V = GF(2f ) and σ(D) = Ds is isomorphic to D.

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Lemma 27 If s ∈ Zn satisfies

(i) gcd(s, n) = 1, (ii) ∀x ∈ GF(2n ) \ {0, 1}, (1 + x)s = 1 + xs , (iii) ∀x = ∀y ∈ GF(2n ) \ {0, 1}, (1 + x + y)s = 1 + xs + y s ,

then the designs Ds and D are isomorphic and disjoint.

Lemma 28 When f is odd, s = 3 satisfies the conditions (i), (ii), (iii) of Lemma 27 Remark:If s satisfies the condition (i), (ii), (iii), then 2i s and s−1 (mod n) also does. When m ≤ 12 is even, s = 1 is the only parameter satisfying the condition (i) and (ii). Table 3. List of s such that D and Ds are disjoint m

# of s

representatives of s

3

2

1, 3

5

6

1, 3, 5, 7, 11, 15

7

12

1, 3, 5, 9, 11, 13, 15, 23, 27, 29, 43, 63

9

14

1, 3, 5, 13, 17, 19, 27, 31, 47, 59, 87, 103, 171, 255

11

24

1, 3, 5, 9, 13, 17, 33, 35, 43, 57, 63, 95, 107, 117, 143, 151, 231, 249, 315, 365, 411, 413, 683, 1023

13

28

1, 3, 5, 9, 13, 17, 33, 57, 65, 67, 71, 127, 171, 191, 241, 287, 347, 367, 635, 723, 911, 1243, 1245, 1453, 1639, 1691, 2731, 4095

′

Assume that s and s′ satisfy the condition of Lemma 27. Then Ds and Ds are also disjoint when s′ s−1 (mod n) satisfies the conditon. By choosing a set S such that s′ s−1 (mod n) satisfies the condition for each s, s′ ∈ S, we obtain a set of disjoint Ds ’s.

Example 29 (i) For f = 5，s = 1, 3, 5, 7, 11, 15 generate six disjoint 3-(25 , 4, 1) designs. (ii) For f = 7，s = 1, 3, 5, 9, 15, 43 generate six disjoint 3-(27 , 4, 1) designs. Hence, we obtain the following t-SEEDs:

Lemma 30 There exists a 3-(25 , 4; df ) SEED containing a 2-(25 , 4; df 2 where df = 2, 6, 6, . . . for f = 3, 5, 7, · · · .

f −1

3

−1

) SEED,

7.5. 5-SEEDs derived from Golay code In this section, we review mutually disjoint 5-designs related to the Golay code and self dual codes. Kramer and Magliveras [21] constructed 9 mutually disjoint Steiner systems S(5, 8, 24) by finding 8 permutations on 24 points. Araya [4] also constructed 15 mutually disjoint Steiner systems S(5, 8, 24) by a computer search. The following results were shown by Jimbo and Shiromoto [18].

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M. Jimbo and K. Shiromoto / Quantum Jump Codes and Related Combinatorial Designs

Theorem 31 There exists at least 22 mutually disjoint Steiner systems S(5, 8, 24). Hence a 5-(24, 8; 22) SEED exists. Theorem 31 can be obtained by making disjoint isomorphic 22 5-(24, 8, 1) designs from the Golay code. We will give a breif proof of Theorem 31. Let G24 be the binary extended Golay [24, 12, 8] code with parity-check matrix ⎛1 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 1 1 0 0 0 1 0 1⎞

⎛ 010000000000011011100011 ⎜ 00 00 10 01 00 00 00 00 00 00 00 00 10 01 10 11 01 10 11 11 01 00 00 11 ⎟ ⎜0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 1 0 1 1 1 0 1⎟ ⎜ ⎟ ⎜ ⎜ H(G24 ) = ⎜ 00 00 00 00 00 10 01 00 00 00 00 00 01 00 00 10 01 10 11 01 10 11 11 11 ⎟ = ⎜ I12 ⎜0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1 0 1 1 0 1 1⎟ ⎝ ⎝0 0 0 0 0 0 0 0 1 0 0 0 1 1 1 0 0 0 1 0 1 1 0 1⎠ 000000000100011100010111 000000000010101110001011 000000000001111111111110

⎞ 1 .⎟ A11 .. ⎟ ⎟. 1⎠ 1 ··· 1 0

Note that A11 is a circulant matrix, and the Hamming distance between any two distinct row vectors of A11 is 6. Let σ = (13, 14, . . . , 23) and τ = (1, 13)(2, 14) · · · (11, 23) be the coordinate permutations which act on the vector space GF(2)24 . We denote the zero vector and the all-one vector by 0 and 1, respectively. For any positive integer m, let Jm be the all-one m × m matrix. The following lemma is well-known and is essential (see, for instance, Ch. 16 in [22]). Lemma 32 Let X be a circulant matrix of first row (c0 , c1 , . . . , cn−1 ) over a finite field. X is invertible if and only if a0 (x) = c0 + c1 x + · · · + cn−1 xn−1 is relatively prime to xn − 1. Now the following three lemmas are obtained. Lemma 33 For any i, j ∈ {0, 1, . . . , 10}, i = j, the intersection between all the σi σj and G24 is {0, 1, x, x + 1}, where x is the weight 12 vector codewords in G24 (0, . . . , 0, 1, . . . , 1, 0). Lemma 34 For any i, j ∈ {0, 1, . . . , 10}, i = j, the intersection between all the τ σi τ σj and G24 is {0, 1, y, y + 1}, where y is the weight 12 vector codewords in G24 (1, . . . , 1, 0, . . . , 0). i

σ τσ Lemma 35 For any i and j, the intersection between all the codewords in G24 and G24 is {0, 1}.

j

By summarizing these results, we have the following: Theorem 36 Let σ = (13, 14, . . . , 23) and τ = (1, 13)(2, 14) · · · (11, 23) be the permutations on 24 points and let H be the set of all permutations of the form τ l σ i in the permutation group S24 . And let B be the set of supports of all the Hamming weight 8 codewords in G24 . Then {B g : g ∈ H} forms the set of 22 mutually disjoint Steiner systems S(5, 8, 24). * In Theorem 36, for any subset K of H, the collection g∈K B g can be viewed as a set of blocks in a simple 5-(24, 8, |K|) design. Then we have the following result as a corollary of Theorem 36.

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Corollary 37 There exist simple 5-(24, 8, m) designs, for m = 1, 2, . . . , 22. It is also known that the set of supports of the codewords of Hamming weight 12 in G24 forms a 5-(24, 12, 48) design. From Proposition 35, there is no codewords of σi τ σj Hamming weight 12 in the intersection between G24 and G24 . Corollary 38 There exists at least two mutually disjoint 5-(24, 12, 48) designs. And there exist simple 5-(24, 12, 48m) designs, for m = 1, 2. Recently, Araya and Harada [5] found the following by a computer search. Theorem 39 (Araya and Harada [5]) There exists at least 50 mutually disjoint Steiner systems S(5, 8, 24). Hence a 5-(24, 8; 50) SEED exists. Theorem 40 (Araya and Harada [5]) There exists at least 35 mutually disjoint 5S(24, 12, 48) designs. Hence a 5-(24, 12; 35) SEED exists. Similar resluts were obtained for a quadratic residue code of length 48. Theorem 41 (Jimbo and Shiromoto [18]) There exists at least 46 mutually disjoint simple 5-(48, 12, 8) designs. Hence a 5-(48, 8; 46) SEED exists. The above results are based on a binary extended Golay code of length 24 and a quadratic residue code of length 48. Angata and Shiromoto [3] and Araya, Harada, Tonchev and Wassermann [6] independently generalized the results to the case of Pless symmetry (ternary) code. Theorem 42 (Angata and Shiromoto [3]) There exist at least (i) 34 mutually disjoint 5-(36, k, λ) designs for each (k, λ) = (12, 45), (15, 5577). (ii) 58 mutually disjoint 5-(60, k, λ) designs, for each (k, λ) = (18, 3060), (21, 449820), (24, 34337160), (27, 1271766600). Remark: Araya, Harada, Tonchev and Wassermann [6] found 17 mutually disjoint 5(36, 12, 45) designs. Theorem 43 (Angata and Shiromoto [3], Araya, Harada, Tonchev, Wassermann [6]) There exist at least 11 mutually disjoint 5-(24, 9, 6) designs. Theorem 44 (Angata and Shiromoto [3]) There exist at least 23 mutually disjoint 5(48, k, λ) designs, for each (k, λ) = (15, 364), (18, 50456), (21, 2957388). By these results, the following is obtained: Corollary 45 There exist (i) (ii) (iii) (iv)

a 5-(36, 12; 34) SEED for k = 12, 15, a 5-(60, k; 58) SEED for k = 18, 21, 24, 27, a 5-(24, 9; 11) SEED, and a 5-(48, k; 23) SEED for k = 15, 18, 21.

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Moreover, Araya, Harada, Tonchev and Wassermann [6] obtained the following. Theorem 46 (Araya, Harada, Tonchev and Wassermann [6]) There exist at least (i) (ii) (iii) (iv) (v) (vi)

3 mutually disjoint 5-(18, 8, 6) designs, 5 mutually disjoint 5-(24, 10, 36) designs, 2 mutually disjoint 5-(25, 9, 30) designs, 2 mutually disjoint 5-(30, 12, 220) designs, 4 mutually disjoint 5-(32, 6, 3) designs. 4 mutually disjoint 5-(33, 7, 4) designs.

Corollary 47 There exist a 5-(18, 8; 2) SEED, a 5-(24, 10; 5) SEED, a 5-(25, 9; 2) SEED, a 5-(30, 12; 2) SEED, a 5-(32, 6; 4) SEED and a 5-(33, 7; 4) SEED. 7.6. More SEEDs from codes By Assmus and Matson [7]’s theorem, codewords of weight k of codes in Table 4 form 3-designs, or 5-designs. If we can partition the design into subdesigns, 3-SEEDs can be obtained. The results in Table 4 were reported by Shiromoto [27]. Table 4. Partition of t-designs derived from codes codes

Extended BCH

Aut(C)

AΓL(1, 32)

[32, 21, 6] Code

It’s dual

AΓL(1, 32)

[32, 11, 12] Code Extended BCH

AGL(2, 5)

[32, 16, 8] Code Self-Dual Extended QR

PSL(2, 31)

[32, 16, 8] Code Self-Dual Extended QR [48, 24, 12] Code

PSL(2, 47)

weights

designs∗1

λ’s of subdesigns

6

3-(32,6,4)

4

8

3-(32,8,119)

56, 56, 7

10

3-(32,10,1464)

120 × 24

12

3-(32,12,10120)

220 × 43, 44, 22 × 3, 110 × 5

14

3-(32,14,32760)

364 × 90

16

3-(32,16,68187)

560 × 119, 112 × 5, 140 × 7, 7

12

3-(32,12,22)

22

16

3-(32,16,119)

7,112

12

3-(32,12,616)

616

16

3-(32,16,4123)

3136, 7, 980

8

3-(32,8,7)

7

12

3-(32,12,616)

11, 165, 110, 330

16

3-(32,16,4123)

112, 336, 560, 840, 210, 140, 105 ,840 ,560 ,420

12

5-(48,12,8)

3-(48,12,λ) 110,55,55

16

5-(48,16,1365)

unknown

20

5-(48,20,36176)

unknown

(*1) Assmus & Matson (1969) (*2) Computations of subdesigns using MAGMA were assisted by M. Angata

From these computation results, the following theorem is shown. Theorem 48 There exist a 3-(32, 8; 3) SEED, a 3-(32, 10; 24) SEED, a 3-(32, 12; 52) SEED, a 3-(32, 14; 90) SEED, and a 3-(32, 16; 132) SEED.

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8. Concluding remark and open problems In this paper, we considered constructions of t-error correcting jump codes and t-SEEDs. Besides the construction of t-SEEDs reviewed in this paper, more constructions are presented in Beth et al. [8] and Charnes and Beth [12]. Beth et al. [8] gave a construction of (n, 2, t)k jump codes by using isodual binary codes, which was extended by Charnes and Beth [12] utilizing a group theoretical technique. However, only a few results are known for optimal t-SEEDs attaining the upperbound (16) for t ≥ 2. In general, t-SEEDs have weaker combinatorial conditions than that of large sets. But we do not know any example of optimal t-SEEDs except for large sets. Problem 49 Is there an optimal t-SEED attaining the upperbound (16) for t ≥ 2 except for large sets? A jump code can be considered as a continuous version of a t-SEED or a system of disjoint t-designs. Actually, “balancedness” is generalized to the constancy of inner product. Whereas, “disjointness” corresponds to orthogonality. It is ovbious that if there is a t-(n, k; m) SEED, then there is a (n, m, t)k jump code. But it may not be known whether there is an example such that there is an (n, m, t)k jump code even if there is no t-(n, k; m) SEED. Problem 50 Is there an (n, m, t)k jump code even if there is no t-(n, k; m) SEED. In paticular, a (7, 3, 2)3 jump code can be constructed by two disjoint 2-(7, 3, 1) designs and one 2-(7, 3, 3) designs. But the upperbound for m is 5. Is there a (7, m, 2)3 jump code for m = 4 or 5? Problem 51 If there is an LS1 (t, k, n) it is optimal in the sense that it attains the upperbound (14). However, in the case when there is no LS1 (t, k, n), can we find an optimal or asymtotically optimal t-(n, k; m) SEED for t ≥ 2? In Subsection 7.5, we showed that there are 22 disjoint 5-(24, 8, 1) designs. Whereas, Harada [16] found 50 disjoint 5-(24, 8, 1) designs by computer search. If an LS1 (5, 8, 24) exists it must have 3 × 17 × 19 disjoint 5-(24, 8, 1) designs. Similarly, we wonder whether a LS48 (5, 12, 24) exists, or not. We pose here a challenging problems. Problem 52 Does there exist an LS1 (5, 8, 24)? Problem 53 Does there exist an LS48 (5, 12, 24)?

Acknowledgements The first author wish to thank the ASI and the organizers, Professor Dean Crnkovi´c and Professor Vladimir Tonchev, for giving me a chance to attend such an extreamly wellorganized conference and for their support.

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References [1] G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl and M. Mussinger, Stabilizing distinguishable qubits against spontaneous decay by detected-jump correcting quantum codes, Physical Review Letters, 86 (2001) 4402–4405. [2] G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl and M. Mussinger, Detected-jump-error-correcting quantum codes, quantum error designs, and quantum computation. Physical Review A, 68 (2003), 012316. [3] M. Angata and K. Shiromoto, Mutually disjoint 5-designs from Pless symmetry codes, submitted to J. Statist. Theory and Practice (2010). [4] M. Araya, More mutually disjoint Steiner systems S(5, 8, 24), J. Combin. Theory, Ser. A 102 (2003), 201–203. [5] M. Araya and M. Harada, Mutually disjoint Steiner systems S(5, 8, 24) and 5-(24, 12, 48) designs, Electronic J. Combinatorics, 17 (2010) N1. [6] M. Araya, M. Harada, V. D. Tonchev, A. Wassermann, Mutually disjoint designs and new 5-designs derived from groups and codes, to appear in J. Combin. Designs, (2010). [7] E. F. Assmus Jr. and H. F. Mattson Jr., (1969), New 5 designs, J. Combin. Theory 6, 122–151. [8] T. Beth, C. Charnes, M. Grassl, G. Alber, A. Delgado, M. Mussinger, A new class of designs which protect against quantum jumps. Designs, Codes and Cryptography, 29 (2003), 51–70. [9] A. R. Calderbank and P. W. Shor, Good quantum error-correcting codes exist, Physical Review A, 54, (2) (1996) 1098–1105. [10] A.R. Calderbank, E.M. Rains, P.W. Shor, N.J.A. Sloane, Quantum error correction and orthogonal geometry, Phys. Rev. Lett., 78 (1997) 405–408. [11] A.R. Calderbank, E.M. Rains, P.W. Shor, N.J.A. Sloane, Quantum error correction via codes over GF(4), IEEE Trans. Inform. Theory 44 (4) (1998) 1369–1387. [12] C. Charnes and T. Beth, Combinatorial aspects of jump codes, Discrete Math., 294 (2005) 43–51. [13] C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs (2nd edition), CRC Press, Boca Raton, 2007. [14] A. Ekert and C. Macchiavello, Quantum error correction for communication, Physical Review Letters, 77, (12) (1996) 2585–2588. [15] M. Grassel, T. Beth and T. Pellizzari, Codes for the quantum erasure channel, Physical Review A, 56 (1997) 33–38. [16] M. Harada, private communication (2010). [17] W. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Pless, 2003. [18] M. Jimbo and K. Shiromoto, A construction of mutually disjoint Steiner systems from isomorphic Golay codes, J. Combin. Theory Ser. A, 116 (2009) 1245–1251. [19] O. Kern and G. Alber, Suppressing decoherence of quantum algorithms by jump codes, European Physical J. D, 36 (2005) 241–248. [20] G.B. Khosrovshahi and B. Tayfeh-Rezaie, Large sets of t-designs through partitionable sets: A survey, Discrete Math., 306 (2006) 2993-3004. [21] E. S. Kramer and S. S. Magliveras, Some mutually disjoint Steiner systems, J. Combin. Theory, Ser. A 17 (1974), 39–43. [22] F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, North-Holland Publishing Company, Amsterdam, 1978. [23] E. Knill and R. Laflamme, Theory of quantum error-correcting codes, Physical Review A, 55 (1997), 900-911. [24] The Magma Computational Algebra System for Algebra, Number Theory and Geometry, Version 2.12, University of Sydney (2005). [25] A. Munemasa, Flag-transitive 2-designs arising from line-spreads in PG(2n-1,2), Geometriae Dedicata, 77 (1999), 209–213. [26] D. Raghavarao, Constructions and Combinatorial Problems in Design of Experiments, Wiley, New York (1971). [27] K. Shiromoto, private communication (2008). [28] P. W. Shor, Scheme for reducing decoherence in quantum computer memory, Physical Review A, 52 (1995) R2493-R2496. [29] A. Steane, Multiple particle interference and quantum error correction, Proceedings of the Royal Society London Series A, 452 (1996) 2551–2577.

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-312

Unbiased Hadamard matrices and bases Hadi Kharaghani 1 Abstract. Mutually unbiased unit Hadamard (MUUH) matrices have been studied for almost 40 years. Recent interest in such matrices has been motivated by their applications to quantum information theory. In this paper, we introduce the class of mutually unbiased complex Hadamard (MUCH) matrices having as sntries fourth roots of unity. The number of MUCH matrices of order 2n, n odd, is at most 2, and the bound is attained for n = 1, 5, 9. Certain pairs of mutually unbiased complex Hadamard matrices of order m can be used to construct pairs of unbiased real Hadamard matrices of order 2m. We then turn our attention to the class of mutually unbiased real Hadamard matrices of order 16n2 which seem to be the most interesting ones. Mutually unbiased weighing (MUW) matrices are introduced for the first time. Further study of these objects look quite promising. Keywords. Complex Hadamard matrix, Hadamard matrix, weighing matrix, unbiased complex Hadamard matrices, mutually unbiased weighing matrices, Mutually suitable Latin squares, MOLS, Bush-type Hadamard matrices, unbiased bases

Introduction A complex (unit) Hadamard matrix is a matrix H of order n with entries in {−1, 1, i, −i} ({c ∈ Cn , |c| = 1}) and orthogonal rows in the usual complex inner product on Cn . If the entries of the matrix consist of only ±1, we call the matrix a real Hadamard matrix. Our main references for complex and real Hadamard matrices are [13,14]. Two complex Hadamard matrices H and K of order 2n are called unbiased if HK ∗ = L, where K ∗ denotes √ the Hermitian transpose of K and all the entries of the matrix L are of the absolute value 2n. In this case, it follows that 2n = a2 + b2 , where a, b are nonnegative integers. While there has been considerable interest in unbiased unit Hadamard matrices, it is only recently that some attention has been given to unbiased real Hadamard matrices subsequent to which interesting applications have emerged [12]. The class of unbiased complex Hadamard matrices was introduced and studied in [2]. We borrow a portion of [2] in this note. The readers are referred to [5,7,15] for the study and applications of unbiased unit Hadamard matrices. The topics studied in this article are as follows. We first concentrate on matrices of order 2n, n odd, with entries in {−1, 1, i, −i}. We will find an upper bound for the number of mutually unbiased complex Hadamard matrices of order 2n, n odd, denoted |MUCH(2n)|. We report on the outcome of a computer search for the classes of MUCH matrices of orders 10 and 18. We then turn to the study of mutually unbiased real Hadamard matrices (MUHM). In the course of studying MUHM, we introduce a class 1 Department of Mathematics & Computer Science, University of Lethbridge, Lethbridge, Alberta, T1K 3M4, Canada; email: [email protected] - supported by an NSERC Discovery Grant - Group.

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of Latin squares which we call mutually suitable Latin squares. It turns out that this class is equivalent to the class of mutually orthogonal Latin squares (MOLS). Next we consider an extension of Hadamard matrices to weighing matrices and touch upon DelsarteGoethals-Seidel [6] and Calderbank-Cameron-Kantor-Seidel bound [4]. We will briefly discuss mutually unbiased bases in the last section. In the presentation of matrices we use j to denote −i and − to denote −1. 1. Unbiased complex Hadamard matrices Dealing with complex Hadamard matrices, i.e. matrices with entries in {−1, 1, i, −i}, is quite different from working with unit Hadamard matrices as the powerful character theory is no longer applicable. We begin this section with a well known [2,9], but important property of complex Hadamard matrices. Lemma 1. Let H = [hi j ] be a complex Hadamard matrix of order n√for which the absolute values of the row sums are all identical and equal to r. Then r = n. Proof. Let e be the all ones column vector, we have (He)∗ (He) = e∗ H ∗ He = e∗ nIe = ne∗ e = n2 . Let ri = ∑nj=1 hi j , 1 ≤ i ≤ n. Noting that He is the column vector with components ri = √ ∑nj=1 hi j and e∗ H ∗ = (He)∗ , we have ∑ni=1 |ri |2 = n2 . It follows that r = n. A complex Hadamard matrix of order n for which the absolute values of the row √ sums are all equal to n is called row regular. It follows from Lemma 1 that for a row regular complex Hadamard matrix H = [hk j ] of order 2n, n odd, if ∑2n j=1 hk j = a + ib, for 2 2 some k, 1 ≤ k ≤ 2n, then a + b = 2n and so both |a| and |b| are odd integers. We use this in the next lemma from [2]. Lemma 2. For the odd integer n, it is not possible to have a pair of unbiased row regular complex Hadamard matrices of order 2n. Proof. Suppose on the contrary that there is a pair of row regular complex Hadamard ∗ matrices √ H and K of order 2n such that HK = L, where the entries of L are of absolute value 2n. Let J be the matrix of all one entries of order 2n. Then the matrix 1 1 (H + J) (K ∗ + J) 1+i 1+i is a complex integer matrix (i.e. all entries of the matrix consist of Gaussian integers). It 1 1 is easy to see that the entries of both matrices 1+i (H + J) and 1+i (K ∗ + J) belong to the set {0, 1, −i, 1 − i}. Noting that 1 −i 1 (H + J) (K ∗ + J) = (HK ∗ + HJ + JK ∗ + 2nJ) 1+i 1+i 2 and that all the entries of the matrices HK ∗ , HJ and JK ∗ consist of numbers of the form x + iy, where both |x| and |y| are odd integers, we get a contradiction.

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Theorem 3. For any odd integer n, |MUCH(2n)| ≤ 2. Proof. Suppose on the contrary that there are more than two MUCH matrices of order 2n. By multiplying the columns of all matrices by appropriate numbers we can make the first row of one of the matrices to be all equal to one. The new matrices form a set of MUCH matrices which contain at least two row regular Hadamard matrices of order 2n, contradicting Lemma 1 and thus the result follows. Example 4. Let H=



11 , −1

K=



1i . i1

Then HK ∗ =



1−i 1−i . −1 − i 1 + i

This shows the inequality in the Theorem 3 is sharp for n = 1. By a computer search many maximal sets of MUCH matrices of orders 10 and 18 were found in [2]. One representative from each of these pairs of matrices is listed below in Tables 1 and 2. Table 1. A pair H, K of unbiased complex Hadamard matrices of order 10 ⎛

⎞ 1111111111 ⎜ 1− i i j i j j j i ⎟ ⎜ ⎟ ⎜ ⎟ ⎜1 i−i j j i i j j ⎟ ⎟ ⎜ ⎜1 i i−i j j j i j ⎟ ⎟ ⎜ ⎟ ⎜ ⎜1 j j i−i i j i j ⎟ ⎟ ⎜ ⎜ 1 i j j i − i j j i ⎟, ⎜ ⎟ ⎜ ⎟ ⎜1 j i j i i−i j j ⎟ ⎜ ⎟ ⎜1 j i j j j i−i i ⎟ ⎜ ⎟ ⎜ ⎟ ⎝1 j j i i j j i−i ⎠ 1 i j j j i j i i−

⎞ j −1 1 1 j i i 1 j ⎜ i 1 i 1 1− j 1 i j ⎟ ⎟ ⎜ ⎟ ⎜ ⎜− j 1 i 1 1 j 1 j i ⎟ ⎜ ⎟ ⎜ 1 i 1 1 i 1 1 j −− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ j 1 i j j 1 i 1−1 ⎟ ⎜ ⎟ ⎜ 1 j −1− j 1 1 1 i ⎟ ⎜ ⎟ ⎜ ⎟ ⎜−1 j 1 i i i j 1 1 ⎟ ⎜ ⎟ ⎜ i i 1 j −1 j i 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1− j −1 i 1 1 i 1 ⎠ ⎛

1 1 1 i j −1− j 1

As stated in [2], it is reasonable to assume that the upper bound in Theorem 3 is sharp for every odd integer n for which 2n is the order of a row regular complex Hadamard matrix. The following conjecture includes this and a conjecture regarding the existence of row regular complex Hadamard matrices. Conjecture 5. |MUCH(2n)| = 2 for all odd integers n, where 2n is a sum of two squares. The existence of a row regular Hadamard matrix is a necessary condition to have two MUCH’s (see the proof of Theorem 3). For matrices of size 2n, n odd, the existence of a row regular Hadamard matrix is, in turn, conditioned by the existence of integers a, b such that 2n = a2 + b2 (see lemma 1).

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Table 2. A pair H, K of unbiased complex Hadamard matrices of order 18 ⎛

111111111111111111

⎜ 1− i i j i j j j i i j j j i j i i ⎜ ⎜ ⎜1 i−i i j i j j j i i j j j i j i ⎜ ⎜1 i i−i i j i j j j i i j j j i j ⎜ ⎜ ⎜1 j i i−i i j i j j j i i j j j i ⎜ ⎜1 i j i i−i i j i j j j i i j j j ⎜ ⎜ ⎜1 j i j i i−i i j i j j j i i j j ⎜ ⎜1 j j i j i i−i i j i j j j i i j ⎜ ⎜ ⎜1 j j j i j i i−i i j i j j j i i ⎜ ⎜1 i j j j i j i i−i i j i j j j i ⎜ ⎜ ⎜1 i i j j j i j i i−i i j i j j j ⎜ ⎜1 j i i j j j i j i i−i i j i j j ⎜ ⎜ ⎜1 j j i i j j j i j i i−i i j i j ⎜ ⎜1 j j j i i j j j i j i i−i i j i ⎜ ⎜ ⎜1 i j j j i i j j j i j i i−i i j ⎜ ⎜1 j i j j j i i j j j i j i i−i i ⎜ ⎜ ⎝1 i j i j j j i i j j j i j i i−i

1 i i j i j j j i i j j j i j i i−

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎛

− 1 1 i 1 j 1 i 1 1 1 −− i i −− 1

⎞

⎜ i i −1 1− i 1−1 i j 1 j i 1−1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i i i i i −1 j 1 1−1−1 1 j 1−⎟ ⎜ ⎟ ⎜ i −1−1 1 j 1− i i i i −1 j 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i 1 1− i 1−1 i j 1 j i 1−1 i −⎟ ⎜ ⎟ ⎜ i 1 1 j − i i 1 1 −− 1 1 i i − j 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ j i j 1 i 1 i i j j i i 1 i 1 j i −⎟ ⎜ ⎟ ⎜ 1 j i i i j 1 i i j −1 1− j i i 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ j i 1 j i−i j 1 i 1 i i j j i i 1⎟ ⎜ ⎟ ⎜ −− 1 i j i 1 − i − 1 1 1 1 1 1 − i ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 i i 1 i −1 j i j i j i j 1− i ⎟ ⎜ ⎟ ⎜ − 1 i 1 1 1 −− i i −− 1 1 1 i 1 j ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ i − i 1−1 1 j j 1 1−1 i − i 1 i ⎟ ⎜ ⎟ ⎜ 1− j j 1 i i i i 1 j j − i 1 i 1 i ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 i i i j 1 i i j −1 1− j i i 1 j ⎟ ⎜ ⎟ ⎜ j 1 j i−i j 1 i 1 i i j j i i 1 i ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1 1− j j j j −1 i i 1 i i i 1 i i ⎠ 1 j −1 i i 1 i i i 1 i i 1− j j j

2. Unbiased real Hadamard matrices of order 4n2 , n odd t Two real Hadamard matrices H, K of order n are √ called unbiased, if HK √ = L, where the absolute values of all entries of L are equal to n. It follows that L = nA, where A is a real Hadamard matrix of order n and so unbiased Hadamard matrices exist only in square orders. Our first lemma is the real version of Lemma 2.

Lemma 6 ([3]). For the odd integer n, it is not possible to have a pair of unbiased row regular Hadamard matrices of order 4n2 . Proof. Repeating the line of proof of Lemma 2, we have 1 1 1 (H + J) (K t + J) = (HK t + HJ + JK t + 4n2 J). 2 2 4 Noting that HK t = 2nL, where L is a Hadamard matrix, we get a contradiction to the fact that the left side of the above identity is an integer matrix. Lemma 7. Let w(n) be the number of mutually unbiased real Hadamard matrices of order 4n2 , n odd, then w(n) ≤ 2. We have shown in [2] that certain pairs of unbiased complex Hadamard matrices can be used to construct pairs of unbiased real Hadamard matrices. Before giving a proof of this we introduce a notation: for the integer a let G(a) = {a ± ia, −a ± ia}. Theorem 8 ([2]). Let H, K be a pair of unbiased complex Hadamard matrices of order 2n, n odd, for which the entries of HK ∗ are all in G(a), where 2n = a2 + a2 , a odd integer. Then there is a pair of unbiased real Hadamard matrices of order 4n.

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Proof. Let H = A + iB, K = C + iD, where A, B and C, D are (0, ±1)-matrices of order 2n such that A ± B and C ± D are ±1-matrices. Consider the matrices



11 −1  H = ⊗A+ ⊗B 1− 11 and 

K =





−1 11 ⊗ D. ⊗C + 11 1−

It is only a routine calculation to see that H  , K  are Hadamard matrices of order 4n. Let HK ∗ = E + iF, where E, F are ±a-matrices of order 2n. We have

2E −2F 2(ACt + BDt ) −2(BCt − ADt ) . = H  K t = 2(BCt − ADt ) 2(ACt + BDt ) 2F 2E Using the fact that the entries of HK ∗ are in G(a) and noting that E, F are (±a)-matrices, it follows that H  , K  are unbiased Hadamard matrices of order 4n. Remark 9. Note that the assumption above that the entries of HK ∗ are all in G(a), where 2n = a2 + a2 , a odd integer, implies that HK ∗ = (a + ia)L, where L is a complex Hadamard matrix. Corollary 10 ([2]). There is a pair of unbiased Hadamard matrices of order 36. Proof. We apply Theorem 8 to the pair of unbiased complex Hadamard matrices of order 18 of Table 2. The resulting pair of matrices is given in Tables 3 and 4. The fact that all entries of HK ∗ are in G(3) is automatic in this case, as 18 = 32 + 32 only.

3. Unbiased Hadamard matrices of order 16n2 We start this section with a characterization of Hadamard matrices. Theorem 11 (Kharaghani [10]). There is a Hadamard matrix of order 2n if and only if there are 2n ±1-matrices C0 , C1 , C2 , . . ., C2n−1 of order 2n such that: 1. 2. 3. 4. 5.

Cit = Ci , CiC j = 0, i = j, Ci2 = 2nCi , C0 +C1 +C2 + · · · +C2n−1 = 2nI2n , C0 may be chosen to be the matrix of all ones.

Proof. Let ri be the (i−1)-th row of the normalized Hadamard matrix H, and let Ci = rit ri , for i = 1, . . . , 2n. Then, 1. Cit = (rit ri )t = Ci . 2. CiC j = rit ri rtj r j = 0, i = j. 3. Ci2 = rit ri rit ri = 2nrit ri = 2nCi .

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Table 3. A pair of unbiased Hadamard matrices of order 36: first matrix ⎛

111111111111111111111111111111111111

⎞

⎜ 1 −−− 1 − 1 1 1 −− 1 1 1 − 1 −− 1 − 1 1 − 1 −−− 1 1 −−− 1 − 1 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ 1 −−−− 1 − 1 1 1 −− 1 1 1 − 1 − 1 1 − 1 1 − 1 −−− 1 1 −−− 1 − 1 ⎟ ⎟ ⎜ ⎜ 1 −−−−− 1 − 1 1 1 −− 1 1 1 − 1 1 1 1 − 1 1 − 1 −−− 1 1 −−− 1 − ⎟ ⎟ ⎜ ⎜ ⎟ ⎜ 1 1 −−−−− 1 − 1 1 1 −− 1 1 1 − 1 − 1 1 − 1 1 − 1 −−− 1 1 −−− 1 ⎟ ⎜ ⎟ ⎜ 1 − 1 −−−−− 1 − 1 1 1 −− 1 1 1 1 1 − 1 1 − 1 1 − 1 −−− 1 1 −−− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 − 1 −−−−− 1 − 1 1 1 −− 1 1 1 − 1 − 1 1 − 1 1 − 1 −−− 1 1 −− ⎟ ⎜ ⎟ ⎜ 1 1 1 − 1 −−−−− 1 − 1 1 1 −− 1 1 −− 1 − 1 1 − 1 1 − 1 −−− 1 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 − 1 −−−−− 1 − 1 1 1 −− 1 −−− 1 − 1 1 − 1 1 − 1 −−− 1 1 ⎟ ⎜ ⎟ ⎜ 1 − 1 1 1 − 1 −−−−− 1 − 1 1 1 − 1 1 −−− 1 − 1 1 − 1 1 − 1 −−− 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 −− 1 1 1 − 1 −−−−− 1 − 1 1 1 1 1 1 −−− 1 − 1 1 − 1 1 − 1 −−− ⎟ ⎜ ⎟ ⎜ 1 1 −− 1 1 1 − 1 −−−−− 1 − 1 1 1 − 1 1 −−− 1 − 1 1 − 1 1 − 1 −− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 −− 1 1 1 − 1 −−−−− 1 − 1 1 −− 1 1 −−− 1 − 1 1 − 1 1 − 1 − ⎟ ⎜ ⎟ ⎜ 1 1 1 1 −− 1 1 1 − 1 −−−−− 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 1 − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 1 1 −− 1 1 1 − 1 −−−−− 1 1 1 −−− 1 1 −−− 1 − 1 1 − 1 1 − ⎟ ⎜ ⎟ ⎜ 1 1 − 1 1 1 −− 1 1 1 − 1 −−−−− 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 − 1 1 1 −− 1 1 1 − 1 −−−− 1 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 ⎟ ⎜ ⎟ ⎜ 1 −− 1 − 1 1 1 −− 1 1 1 − 1 −−− 1 1 1 − 1 −−− 1 1 −−− 1 − 1 1 − ⎟ ⎜ ⎟ H =⎜ ⎟ ⎜ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −−−−−−−−−−−−−−−−−− ⎟ ⎜ ⎟ ⎜ 1 − 1 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 1 1 − 1 −−− 1 1 −−− 1 − 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 − 1 1 − 1 −−− 1 1 −−− 1 − 1 − 1 1 1 1 − 1 −−− 1 1 −−− 1 − 1 ⎟ ⎜ ⎟ ⎜ 1 1 1 − 1 1 − 1 −−− 1 1 −−− 1 −− 1 1 1 1 1 − 1 −−− 1 1 −−− 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 1 − 1 1 − 1 −−− 1 1 −−− 1 −− 1 1 1 1 1 − 1 −−− 1 1 −−− 1 ⎟ ⎜ ⎟ ⎜ 1 1 − 1 1 − 1 1 − 1 −−− 1 1 −−−− 1 − 1 1 1 1 1 − 1 −−− 1 1 −−− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 − 1 1 − 1 1 − 1 −−− 1 1 −−−− 1 − 1 1 1 1 1 − 1 −−− 1 1 −− ⎟ ⎜ ⎟ ⎜ 1 −− 1 − 1 1 − 1 1 − 1 −−− 1 1 −−−− 1 − 1 1 1 1 1 − 1 −−− 1 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 −−− 1 − 1 1 − 1 1 − 1 −−− 1 1 −−−− 1 − 1 1 1 1 1 − 1 −−− 1 1 ⎟ ⎜ ⎟ ⎜ 1 1 −−− 1 − 1 1 − 1 1 − 1 −−− 1 − 1 −−− 1 − 1 1 1 1 1 − 1 −−− 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 −−− 1 − 1 1 − 1 1 − 1 −−−− 1 1 −−− 1 − 1 1 1 1 1 − 1 −−− ⎟ ⎜ ⎟ ⎜ 1 − 1 1 −−− 1 − 1 1 − 1 1 − 1 −−−− 1 1 −−− 1 − 1 1 1 1 1 − 1 −− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 −− 1 1 −−− 1 − 1 1 − 1 1 − 1 −−−− 1 1 −−− 1 − 1 1 1 1 1 − 1 − ⎟ ⎜ ⎟ ⎜ 1 −−− 1 1 −−− 1 − 1 1 − 1 1 − 1 −−−− 1 1 −−− 1 − 1 1 1 1 1 − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 −−− 1 1 −−− 1 − 1 1 − 1 1 −− 1 −−− 1 1 −−− 1 − 1 1 1 1 1 − ⎟ ⎜ ⎟ ⎜ 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 1 −− 1 −−− 1 1 −−− 1 − 1 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1 1 − 1 −−− 1 1 −−− 1 − 1 1 − 1 − 1 − 1 −−− 1 1 −−− 1 − 1 1 1 1 ⎠ 1 1 1 − 1 −−− 1 1 −−− 1 − 1 1 −− 1 1 − 1 −−− 1 1 −−− 1 − 1 1 1

4. This follows from the fact that 

⎛

⎜ ⎜ t r0t r1t . . . r2n−1 ⎜ ⎝

r0 r1 .. . r2n−1

⎞

⎟ ⎟ ⎟ = 2nI2n . ⎠

5. Note that the first row consist of all one entries.

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Table 4. A pair of unbiased Hadamard matrices of order 36: second matrix ⎛

− 1 1 − 1 1 1 − 1 1 1 −−−−−− 1 − 1 1 1 1 − 1 1 1 1 1 −− 1 1 −− 1

⎞

⎜ −−− 1 1 −− 1 − 1 − 1 1 1 − 1 − 1 1 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 − 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜ −−−−−− 1 1 1 1 − 1 − 1 1 1 1 − 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 − 1 − ⎟ ⎟ ⎜ ⎜ −− 1 − 1 1 1 1 −−−−−− 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 1 1 − 1 − 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 1 1 −− 1 − 1 − 1 1 1 − 1 − 1 −− 1 1 1 − 1 1 − 1 1 − 1 − 1 1 − 1 1 − ⎟ ⎜ ⎟ ⎜ − 1 1 1 −−− 1 1 −− 1 1 −−− 1 1 1 1 1 −− 1 1 1 1 −− 1 1 1 1 −− 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 1 − 1 −− 1 1 −− 1 − 1 1 −−− 1 − 1 1 1 1 1 −− 1 1 1 1 1 − 1 − ⎟ ⎜ ⎟ ⎜ 1 1 −−− 1 1 −− 1 − 1 1 − 1 −− 1 1 − 1 1 1 − 1 1 1 −− 1 1 −− 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 1 −−− 1 1 − 1 −− 1 1 −− 1 − 1 1 − 1 − 1 − 1 1 1 1 1 −− 1 1 1 ⎟ ⎜ ⎟ ⎜ −− 1 − 1 − 1 −−− 1 1 1 1 1 1 −−−− 1 1 − 1 1 − 1 − 1 1 1 1 1 1 − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 −− 1 −− 1 1 − 1 − 1 − 1 1 −− 1 1 1 1 1 1 − 1 − 1 − 1 − 1 − 1 − 1 ⎟ ⎜ ⎟ ⎜ − 1 − 1 1 1 −−−−−− 1 1 1 − 1 1 − 1 1 1 1 1 −− 1 1 −− 1 1 1 1 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ −−− 1 − 1 1 1 1 1 1 − 1 −−− 1 − 1 − 1 1 − 1 1 −− 1 1 − 1 1 − 1 1 1 ⎟ ⎜ ⎟ ⎜ 1 − 1 1 1 −−−− 1 1 1 −− 1 − 1 − 1 −−− 1 1 1 1 1 1 −−− 1 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 −−− 1 1 −− 1 − 1 1 − 1 −− 1 1 1 1 1 1 − 1 1 1 −− 1 1 −− 1 1 1 − ⎟ ⎜ ⎟ ⎜ 1 1 1 −−− 1 1 − 1 −− 1 1 −− 1 −− 1 − 1 − 1 − 1 1 1 1 1 −− 1 1 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 − 1 1 1 1 − 1 −− 1 −−− 1 −− 1 1 −−−−−− 1 1 1 1 1 1 1 1 1 1 ⎟ ⎜ ⎟ ⎜ 1 1 − 1 −− 1 −−− 1 −− 1 − 1 1 1 1 −− 1 1 1 1 1 1 1 1 1 1 1 −−−− ⎟ ⎜ ⎟ K=⎜ ⎟ ⎜ − 1 1 1 1 − 1 1 1 1 1 −− 1 1 −− 1 1 −− 1 −−− 1 −−− 1 1 1 1 1 1 − ⎟ ⎜ ⎟ ⎜ 1 1 − 1 1 − 1 1 − 1 1 − 1 − 1 1 − 1 1 1 1 −− 1 1 − 1 − 1 −−− 1 − 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 1 − 1 − 1 1 − 1 − 1 1 − 1 − 1 1 1 1 1 1 −−−− 1 − 1 −−−− 1 ⎟ ⎜ ⎟ ⎜ 1 − 1 − 1 1 − 1 − 1 1 1 1 − 1 − 1 1 1 1 − 1 −−−− 1 1 1 1 1 1 −−−− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 − 1 1 − 1 1 − 1 − 1 1 − 1 1 − 1 −− 1 1 − 1 − 1 −−− 1 − 1 − 1 1 ⎟ ⎜ ⎟ ⎜ 1 1 1 −− 1 1 1 1 −− 1 1 1 1 −− 1 1 −−− 1 1 1 −− 1 1 −− 1 1 1 −− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 1 − 1 1 1 1 1 −− 1 1 1 1 1 − 1 −− 1 −− 1 − 1 1 −− 1 1 − 1 −− 1 1 ⎟ ⎜ ⎟ ⎜ 1 − 1 1 1 − 1 1 1 −− 1 1 −− 1 1 1 −− 1 1 1 −− 1 1 − 1 −− 1 − 1 1 − ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ − 1 1 − 1 − 1 − 1 1 1 1 1 −− 1 1 1 − 1 −− 1 1 1 −− 1 − 1 1 −− 1 1 − ⎟ ⎜ ⎟ ⎜ −− 1 1 − 1 1 − 1 − 1 1 1 1 1 1 − 1 1 1 − 1 − 1 − 1 1 1 −−−−−− 1 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 1 1 − 1 − 1 − 1 − 1 − 1 − 1 −− 1 1 − 1 1 −− 1 − 1 − 1 −− 1 1 ⎟ ⎜ ⎟ ⎜ − 1 1 1 1 1 −− 1 1 −− 1 1 1 1 1 − 1 − 1 −−− 1 1 1 1 1 1 −−− 1 −− ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 − 1 1 − 1 1 −− 1 1 − 1 1 − 1 1 1 1 1 1 − 1 −−−−−− 1 − 1 1 1 − 1 ⎟ ⎜ ⎟ ⎜ 1 −−− 1 1 1 1 1 1 −−− 1 1 1 1 1 − 1 −−− 1 1 1 1 −−− 1 1 − 1 − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 1 1 1 1 − 1 1 1 −− 1 1 −− 1 1 1 −− 1 1 1 −− 1 1 − 1 −− 1 − 1 1 −− ⎟ ⎜ ⎟ ⎜ − 1 − 1 − 1 − 1 1 1 1 1 −− 1 1 1 1 −−− 1 1 1 −− 1 − 1 1 −− 1 1 − 1 ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 1 1 −−−−−− 1 1 1 1 1 1 1 1 1 1 −− 1 −−−− 1 − 1 1 − 1 1 1 − 1 1 ⎠ 1 −− 1 1 1 1 1 1 1 1 1 1 1 −−−−−− 1 − 1 1 − 1 1 1 − 1 1 − 1 −−−

Conversely, the existence of a Hadamard matrix of order 2n follows from the property 2 by selecting one row from each of the Ci s and forming a matrix of order 2n. The first proof of the following theorem traces back to Delsarte, Goethals and Seidel [6] (see [3]). Theorem 12. Let U = {H1 , H2 , · · · , Hm } be a set of mutually unbiased real (respectively complex) Hadamard matrices of order 2n. Then m ≤ n.

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Proof. For 1 ≤ j ≤ m, let C1 j ,C2 j , . . . ,C(2n) j , be the matrices corresponding to the Hadamard matrix H j by applying Theorem 11. Let S j = {Ci j | 1 ≤ i ≤ 2n − 1}. Then the span of each of {S j | 1 ≤ j ≤ 2n} is a subspace of all symmetric matrices of order 2n with zero diagonal of dimension 2n − 1. Using the assumption that the Hadamard matrices are mutually unbiased, it can be seen that every pair of matrices in different subspaces are orthogonal in the inner product defined by A, B = trace(AB∗ ) for square matrices A, B. The span of all Si ’s is contained in the set of all symmetric (respectively Hermitian) matrices with zero diagonal. So m(2n − 1) ≤ (1 + 2 + · · · + 2n − 1) = n(2n − 1) (respectively m(2n − 1) ≤ 2(1 + 2 + · · · + 2n − 1) = 2n(2n − 1)). This completes the proof. Remark 13. The upper bound in previous theorem is attained for real Hadamard matrices of order 4k , see [5]. Next we introduce a class of Latin squares which seemed to have appeared in [8] for the first time. Definition 14. Two Latin squares L1 and L2 of size n on the symbol set {0, 1, 2, . . . , n − 1} are called suitable if every superimposition of each row of L1 on each row of L2 results in only one element of the form (a, a). Example 15. The following are three mutually suitable Latin squares of size 4: ⎛

⎞ 0231 ⎜2 0 1 3⎟ ⎜ ⎟ ⎝3 1 0 2⎠, 1320

⎛

⎞ 0312 ⎜3 0 2 1⎟ ⎜ ⎟ ⎝1 2 0 3⎠, 2130

⎛

⎞ 0123 ⎜1 0 3 2⎟ ⎜ ⎟ ⎝2 3 0 1⎠. 3210

It turns out that from a pair of mutually orthogonal Latin squares (see [1]), one can construct a pair of mutually suitable Latin square and vice versa. Lemma 16 ([8]). There are m MOLS (Mutually Orthogonal Latin Squares) of size n if and only if there are m MSLS (Mutually Suitable Latin Squares) of size n. Proof. Let L1 , L2 be two orthogonal Latin squares on {1, 2, · · · , n} both having their row and columns labeled by the set. Let ((i, j), k) denote the entry at (i, j) position of a Latin square. Then the transformation ((i, j), k) → ((k, j), i) results in a pair of suitable Latin squares. The reverse implication is now clear. Lemma 17 ([1]). Let q be a prime power. Then there are q − 1 MSLS of size q. The class of Bush-type Hadamard matrices which is defined next, and was introduced in [10], is proved to be one of the most versatile class of Hadamard matrices, see [11]. Definition 18. A Bush-type Hadamard matrix is a block matrix H = [Hi j ] of order 4n2 with block size 2n, Hii = J2n and Hi j J2n = J2n Hi j = 0, i = j, 1 ≤ i ≤ 2n, 1 ≤ j ≤ 2n where J2n is the 2n by 2n matrix of all 1 entries. Example 19. Let

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⎞ 11 1 1 ⎜1 1 − −⎟ ⎟ H4 = ⎜ ⎝1 − 1 −⎠. 1−− 1 ⎛

The matrices corresponding to this Hadamard matrix are: ⎞ ⎞ ⎛ ⎛ 1 −− 1 1111 ⎜− 1 1 −⎟ ⎜1 1 1 1⎟ ⎟ ⎟ C1 = r1t r1 = ⎜ C0 = r0t r0 = ⎜ ⎝− 1 1 −⎠, ⎝1 1 1 1⎠, 1111 1 −− 1 ⎞ ⎞ ⎛ ⎛ 1−1− 1 1 −− ⎜− 1 − 1 ⎟ ⎜ 1 1 − −⎟ t ⎟ ⎟ ⎜ C2 = r2t r2 = ⎜ ⎝ 1 − 1 − ⎠ , C3 = r3 r3 = ⎝ − − 1 1 ⎠ . −1−1 −− 1 1 The matrix

⎛

⎞ C0 C1 C2 C3 ⎜ C1 C0 C3 C2 ⎟ ⎟ L=⎜ ⎝ C2 C3 C0 C1 ⎠ C3 C2 C1 C0

is a Bush-type Hadamard matrix of order 16.

We are now ready to construct a very interesting class of mutually unbiased real Hadamard matrices. Theorem 20 ([8]). If there are m MSLS of size 2n, where 2n is the order of a Hadamard matrix, then there are m mutually unbiased Bush-type Hadamard matrices of order 4n2 . Proof. Let C0 ,C1 , . . . ,C2n−1 , be the matrices corresponding to the normalized Hadamard matrix of order 2n. We can assume that all Latin squares are on the set {0, 1, · · · , 2n − 1} and their row and columns are all labeled by the set. Replace the entry i in each of the Latin squares by the matrix Ci , 0 ≤ i ≤ 2n − 1 would result in m mutually unbiased Bush-type Hadamard matrices of order 4n2 . Example 21 ([8]). Let ⎛

⎞ C0 C2 C3 C1 ⎜ C2 C0 C1 C3 ⎟ ⎟ H1 = ⎜ ⎝ C3 C1 C0 C2 ⎠ , C1 C3 C2 C0 ⎛

⎛

⎞ C0 C3 C1 C2 ⎜ C3 C0 C2 C1 ⎟ ⎟ H2 = ⎜ ⎝ C1 C2 C0 C3 ⎠ , C2 C1 C3 C0

⎞ C0 C1 C2 C3 ⎜ C1 C0 C3 C2 ⎟ ⎟ H3 = ⎜ ⎝ C2 C3 C0 C1 ⎠ . C3 C2 C1 C0

and

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These are remarkable matrices. The three matrices are symmetric and -

1 1 1 H1 , H2 , H3 , I16 4 4 4

.

forms a group under matrix multiplication. The blocks are not sign sensitive, i.e., one can change the block signs without changing the unbiasedness of the matrices. Corollary 22. There are 2n − 1 mutually unbiased Bush-type Hadamard matrices of order 22n , n ≥ 2. Proof. This follows from Lemma 17, Theorem 20, and the existence of Hadamard matrices of order 2n for any positive integer n. Remark 23. The unbiased Bush-type Hadamard matrices constructed above can be easily complemented by a normalized Hadamard matrix (see 25 below). However, by doing so we get a maximal set of MUH matrices as we see in the next Lemma. A vector v of dimension m is called unbiased with the√matrix K of order m if the inner product of v with every row of K is of absolute value m. Lemma 24. If there is a normalized Hadamard matrix H which is unbiased with all Bush-type Hadamard matrices of corollary 22. Then there is no ±1-vector unbiased with H and all of the Bush-type Hadamard matrices of corollary 22. Proof. Let 2n = m and v = (e1 , e2 , . . . , em ), where ei s are ±1-vectors of dimension m with the first component ai . Assuming that v is unbiased with all Bush-type Hadamard matrices and H, we get ms(e1 ) + m(a2 + a3 + . . . + am ) = (b1 + b2 + . . . + bm )m, where s(e1 ) denotes the sum of the components of e1 and bi ∈ {−1, 1}. Noting that s(e1 ) and b1 + b2 + . . . + bm are even integers, we get a contradiction to the fact that a2 + a3 + . . . + am is an odd integer. Next we give a lower bound for the number of MUH matrices. Theorem 25 ([8]). Let m be the number of mutually suitable Latin squares of size 2n, where 2n is the order of a Hadamard matrix H, then there are m + 1 mutually unbiased Hadamard matrices of order 4n2 . Proof. Let ri be the i-th row of H, and let K be the block matrix defined by K = [ki j ] = [rtj ri ], i, j = 0, 1, · · · , 2n − 1. It is easy to see that K is a Hadamard matrix of order 4n2 which is unbiased with all the Bush-type Hadamard matrices constructed in Theorem 20. The lower bound in Theorem 25 has appeared in a number of papers, see for example [3,12]. Next we show that our method above extends to weighing matrices.

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4. Extension to weighing matrices One of the advantages of the construction method above is that it can be easily applied to weighing matrices. A matrix W = [wi j ] of order n and wi j ∈ {−1, 0, 1} is called a weighing matrix with weight p and denoted by W (n, p), if WW t = pIn , where In is the identity matrix of order n, see [13]. Two weighing matrices W1 ,W2 of order n and weight √ p are called unbiased, if W1W2t = pW , where W is a weighing matrix of order n and weight p. Theorem 26. Let m be the number of mutually suitable Latin squares of size n, where n is the order of a weighing matrix W with weight p, then there are m + 1 mutually unbiased weighing matrices W (n2 , p2 ). Proof. Note that every step in Theorem 25 can be applied to the weighing matrices and the result is immediate. Example 27. Let ⎞ 1 0 0 − 0 −− ⎜−1 0 0−0−⎟ ⎟ ⎜ ⎜ −− 1 0 0 − 0 ⎟ ⎟ ⎜ ⎟ W =⎜ ⎜ 0 −− 1 0 0 − ⎟ . ⎜ − 0 −− 1 0 0 ⎟ ⎟ ⎜ ⎝ 0 − 0 −− 1 0 ⎠ 0 0 − 0 −− 1 ⎛

The matrices from Theorem 11 corresponding to this weighing matrix are: ⎞ ⎞ ⎞ ⎛ ⎛ 1−0 0 1 0 1 1 0 0 − 0 −− 1 1−0 0 1 0 ⎜−1 0 0−0−⎟ ⎜ 1 1−0 0 1 0 ⎟ ⎜0000000⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜0000000⎟ ⎜ −− 1 0 0 − 0 ⎟ ⎜0000000⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ C0 = ⎜ ⎜ − 0 0 1 0 1 1 ⎟ , C1 = ⎜ 0 0 0 0 0 0 0 ⎟ , C2 = ⎜ 0 0 0 0 0 0 0 ⎟ , ⎜ 1−0 0 1 0 1 ⎟ ⎜0000000⎟ ⎜0000000⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎝0000000⎠ ⎝−0 0 1 0 1 1 ⎠ ⎝ 1 1−0 0 1 0 ⎠ 1−0 0 1 0 1 0000000 −0 0 1 0 1 1 ⎞ ⎞ ⎞ ⎛ ⎛ ⎛ 0000000 1 0 1 1−0 0 0000000 ⎜ 0 1 0 1 1−0 ⎟ ⎜0000000⎟ ⎜ 0 1 1−0 0 1 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎜0000000⎟ ⎜ 1 0 1 1−0 0 ⎟ ⎜ 0 1 1−0 0 1 ⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎟ ⎟ ⎟ ⎜ ⎜ C3 = ⎜ ⎜ 0 −− 1 0 0 − ⎟ , C4 = ⎜ 1 0 1 1 − 0 0 ⎟ , C5 = ⎜ 0 1 0 1 1 − 0 ⎟ , ⎜ 0 1 0 1 1−0 ⎟ ⎜ − 0 −− 1 0 0 ⎟ ⎜0000000⎟ ⎟ ⎟ ⎟ ⎜ ⎜ ⎜ ⎝ 0 − 0 −− 1 0 ⎠ ⎝0000000⎠ ⎝0000000⎠ 0000000 0 1 1−0 0 1 0000000 ⎛

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⎛

323

⎞

0000000 ⎜0000000⎟ ⎜ ⎟ ⎜ 0 0 1 0 1 1−⎟ ⎟ ⎜ ⎟ C6 = ⎜ ⎜ 0 0 0 0 0 0 0 ⎟. ⎜ 0 0 1 0 1 1−⎟ ⎜ ⎟ ⎝ 0 0 1 0 1 1−⎠ 0 0 − 0 −− 1 Substituting these in the appropriate MSLS, we get the following six mutually unbiased weighing matrices: ⎛

⎞ C0C5C3C1C6C4C2 ⎜ C2C0C5C3C1C6C4 ⎟ ⎜ ⎟ ⎜ C4C2C0C5C3C1C6 ⎟ ⎜ ⎟ ⎟ W2 = ⎜ ⎜ C6C4C2C0C5C3C1 ⎟ , ⎜ C1C6C4C2C0C5C3 ⎟ ⎜ ⎟ ⎝ C3C1C6C4C2C0C5 ⎠ C5C3C1C6C4C2C0

⎛

⎞ C0C1C2C3C4C5C6 ⎜ C6C0C1C2C3C4C5 ⎟ ⎜ ⎟ ⎜ C5C6C0C1C2C3C4 ⎟ ⎜ ⎟ ⎟ W3 = ⎜ ⎜ C4C5C6C0C1C2C3 ⎟ , ⎜ C3C4C5C6C0C1C2 ⎟ ⎜ ⎟ ⎝ C2C3C4C5C6C0C1 ⎠ C1C2C3C4C5C6C0

⎞ C0C3C6C2C5C1C4 ⎜ C4C0C3C6C2C5C1 ⎟ ⎟ ⎜ ⎜ C1C4C0C3C6C2C5 ⎟ ⎟ ⎜ ⎟ W4 = ⎜ ⎜ C5C1C4C0C3C6C2 ⎟ , ⎜ C2C5C1C4C0C3C6 ⎟ ⎜ ⎟ ⎝ C6C2C5C1C4C0C3 ⎠ C3C6C2C5C1C4C0

⎞ C0C2C4C6C1C3C5 ⎜ C5C0C2C4C6C1C3 ⎟ ⎜ ⎟ ⎜ C3C5C0C2C4C6C1 ⎟ ⎜ ⎟ ⎟ W5 = ⎜ ⎜ C1C3C5C0C2C4C6 ⎟ , ⎜ C6C1C3C5C0C2C4 ⎟ ⎜ ⎟ ⎝ C4C6C1C3C5C0C2 ⎠ C2C4C6C1C3C5C0

⎞ C0C6C5C4C3C2C1 ⎜ C1C0C6C5C4C3C2 ⎟ ⎟ ⎜ ⎜ C2C1C0C6C5C4C3 ⎟ ⎟ ⎜ ⎟ W6 = ⎜ ⎜ C3C2C1C0C6C5C4 ⎟ , ⎜ C4C3C2C1C0C6C5 ⎟ ⎜ ⎟ ⎝ C5C4C3C2C1C0C6 ⎠ C6C5C4C3C2C1C0

⎞ C0C4C1C5C2C6C3 ⎜ C3C0C4C1C5C2C6 ⎟ ⎜ ⎟ ⎜ C6C3C0C4C1C5C2 ⎟ ⎜ ⎟ ⎟ W1 = ⎜ ⎜ C2C6C3C0C4C1C5 ⎟ , ⎜ C5C2C6C3C0C4C1 ⎟ ⎜ ⎟ ⎝ C1C5C2C6C3C0C4 ⎠ C4C1C5C2C6C3C0

⎛

⎛

⎛

⎛

Delsarte, Goethals and Seidel, [6] studied lines in both Rn and Cn having a prescribed number of angles and found a number of upper bounds, depending on the angles. Calderbank, Cameron, Kantor and Seidel, [4], among other interesting results, found an upper bound for the number of lines in Rn (Cn ) that are either perpendicular or at a fixed angle θ. Having a set of unbiased weighing matrices of order n provides a set of lines in Rn which are either perpendicular or at a fixed angle θ. As it is demonstrated in [6] and [4], line sets meeting the upper bound have very nice properties.

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Theorem 28 ([6,4]). Let m be the cardinality of a set of vectors in Rn such that the absolute value of the inner product of any distinct pair of elements is in {0, α}, 0 < α < 1. Then m≤

n(n + 2)(1 − α2 ) , 3 − (n + 2)α2

assuming that the denominator is positive. If equality holds, then perpendicularity defines a strongly regular graph. Computer computational works show that for some small values of n, the upper bound is attained by unbiased weighing matrices. This is a new area of research and looks quite promising.

5. Unbiased bases Let H, K be a pair of special unbiased complex Hadamard matrices of order 2n2 corresponding to the decomposition 2n2 = n2 + n2 , so that HK ∗ = (n + in)L, for some complex Hadamard matrix L. Then the normalized rows of H and K, or equivalently the rows 2 of √ 1 2 H and √ 1 2 K, form two orthonormal bases for C2n in such a way that for ev2n

2n

1 1 1 (1 + i), − 2n (1 + i), 2n (1 − ery pair of vectors u, v from different bases, u, v ∈ D = { 2n 1 1 n+in i), − 2n (1 − i)} (note that 2n2 = 2n (1 + i)). Here ,  denotes the standard Hermitian in2

√ b : b ∈ Bs }, where Bs denotes the standard basis in ner product in C2n . Adding { 1+i 2 2

2

C2n , to these bases we get 3 orthonormal bases for C2n in such a way that for every pair of vectors u, v from different bases, u, v ∈ D . Two orthonormal bases B1 and B2 in 2 C2n are called unbiased complex bases if u, v ∈ D for all u ∈ B1 and v ∈ B2 . We will use |MUCB(n)| to denote the number of elements in a set of mutually unbiased complex bases for Cn . Lemma 29. |MUCB(2n2 )| ≤ 3 for any odd integer n. Equality is attained for n = 1, 3. 2

Proof. Let B1 , B2 , B3 be three mutually unbiased complex bases for C2n . Let Hi be the 2n H2 H1∗ matrix formed by putting the vectors of Bi as the rows of Hi , i = 1, 2, 3. Then 1+i 2n and 1+i H3 H1∗ form a special pair of unbiased complex Hadamard matrices of order 2n2 corresponding to the decomposition 2n2 = n2 + n2 . Thus, it follows from Theorem 3 that |MUCB(2n2 )| − 1 ≤ 2. The equality occurs for n = 1, 3 as there are pair of special unbiased complex Hadamard matrices of order 2 and 18. Two orthonormal bases B1 and B2 for Rn are called mutually unbiased real bases if u, v ∈ { √1n , − √1n } for all u ∈ B1 and v ∈ B2 , where ,  is the standard Euclidean inner product in Rn , see [3] for details. We will use |MURB(n)| to denote the number of elements in a set of mutually unbiased real bases in Rn . Lemma 30. |MURB(4n2 )| ≤ 3 for any odd integer n. Equality is attained for n = 1, 3.

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4n2

Proof. Let B1 , B2 , B3 be three mutually unbiased real bases for R . Let Hi be the matrix formed by putting the vectors of Bi as the rows of Hi , i = 1, 2, 3. Then 2nH2 H1t and 2nH3 H1t form a pair of unbiased Hadamard matrices of order 4n2 . The result now follows from Lemma 7 and Corollary 10. See also Observation 2.1 of [3]. 2

For the unbiased bases in R16n , following the idea above, it is easy to get a lower bound depending on the number of known MOLS. Using Theorem 12, we see that the 2 upper bound for the number of unbiased bases in R16n is 8n2 + 1. This upper bound is believed to be quite large and very unlikely achievable, except of course, for the case where the dimension of the space is a power of 4. See [3,15] for details. Acknowledgments: This article is based on a joint work of the author with W. Holzmann and W. Orrick [8] and another joint work with Darcy Best [2].

References [1] R. Julian R. Abel, Charles Colbourn, Jeffrey Dinitz, Mutually Orthogonal Latin Squares (MOLS), in Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.), Second Edition, pp. 160–193, Chapman & Hall/CRC Press, Boca Raton, FL, 2007. [2] Darcy Best, H. Kharaghani, Unbiased complex Hadamard matrices and bases, Cryptography and Communications - Discrete Structures, Boolean Functions and Sequences, to appear. [3] P. O. Boykin, M. Sitharam, M. Tarifi and P. Wocjan, Real mutually unbiased bases. Preprint. arXiv:quant ph/0502024v2 [math.CO], (revised version dated Feb. 1, 2008). [4] A. R. Calderbank, P. J. Cameron, W. M. Kantor, J. J. Seidel, Z4 -Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets, Proc. London Math. Soc. 75 (1997), 436–480. [5] P. J. Cameron and J. J. Seidel, Quadratic forms over GF(2), Nederl. Akad. Wetensch. Proc. Ser. A 76=Indag. Math, 35 (1973), 1–8. [6] P. Delsarte, J. M. Goethals, and J. J. Seidel, Bounds for systems of lines and Jacobi polynomials, Philips Res. Repts., 30 (1075), 91–105. [7] Chris Godsil, Aidan Roy, Equiangular lines, mutually unbiased bases, and spin models, European J. Combin., 30 (2009), 246–262. [8] W. Holzmann, H. Kharaghani and W. Orrick, On real unbiased Hadamard matrices, to appear. [9] H. Kharaghani, Jennifer Seberry, The excess of complex Hadamard matrices, Graphs Combin. 9 (1993), 47–56. [10] H. Kharaghani, New class of weighing matrices, Ars Combin., 19 (1985), 69–72. [11] H. Kharaghani, On the twin designs with the Ionin-type parameters, Electron. J. Combin. 7 (2000), Research Paper 1, 11 pp. [12] Nicholas LeCompte, William J. Martin, William Owens, On the equivalence between real mutually unbiased bases and a certain class of association schemes, European J. Combin., to appear. [13] J. Seberry and M. Yamada, Hadamard matrices, sequences, and block designs, in Contemporary Design Theory: A Collection of Surveys, J. H. Dinitz and D. R. Stinson, eds., John Wiley Sons, Inc., 1992, pp. 431–560. [14] Wojciech Tadej, Karol Zyczkowski, A concise guide to complex Hadamard matrices, Open Syst. Inf. Dyn. 13 (2006), 133–177. [15] P. Wocjan and T. Beth, New construction of mutually unbiased bases in square dimensions, Quantum Inf. Comput. 5 (2005), 93–101.

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-326

Multi-structured designs and their applications Ryoh FUJI-HARA1 and Ying MIAO Graduate School of Systems and Information Engineering University of Tsukuba Tsukuba 305-8573, Japan Abstract. In this paper, we give a survey of multi-structured designs and their various applications. Multi-structured designs are block designs with additional structure imposed on the blocks. Combinatorial conditions of the further block structure depend on applications. Examples of multi-structured designs are nested designs, row-column designs, splitting design, etc. Their constructions have been independently studied for a long time and often use similar techniques. Cyclic multistructured designs are related to various types of sequences which have many applications to modern communications. The paper consists of three parts. The first part deals with classical multi-structured designs which are used in experimental designs and authentication systems. The second part reviews several applications to optical and wireless communications which use cyclic multi-structured designs. Finally, in the third part, we discuss algebraic and geometric construction methods which are commonly used for many types of multi-structured designs. Keywords. multi-structured design, experimental design, nested design, balanced array, row-column design, authentication code, optical orthogonal code, frequency hopping sequence, ultra-wideband, comma-free code, difference system of sets, algebraic construction, geometric construction.

1. Introduction Let V be a set of v elements and B a collection of subsets of V . The elements of V and B are called points and blocks, respectively, and the pair (V, B) is called a design. There are many types of designs, for example, the well-known pairwise balanced designs, (r, λ)designs, group divisible designs, and balanced incomplete block designs, each satisfying certain specific combinatorial conditions. The commonly required combinatorial conditions are parts or the whole of the following: (C1) every block contains k points (called regular condition), (C2) every point of V is contained in r blocks (called singleton balance condition) (C3) every pair of distinct points of V appears in exactly in λ blocks (called pair balance condition). 1 Corresponding author. E-mail: [email protected]

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A design (V, B) is called a pairwise balanced design (PBD) if it satisfies the condition (C3), and an (r, λ)-design if it satisfies further the condition (C2). If all the conditions (C1), (C2) and (C3) are satisfied, then (V, B) is said to be a balanced incomplete block design (BIBD) or 2-design, denoted by B(k, λ; v). The reader is referred to [2] for the detailed information on various types of designs. Usually, people don’t consider the fine structures within each block of B. Nevertheless, people also found it necessary sometimes to investigate those designs where some additional combinatorial conditions are imposed on the blocks. A multi-structured design (MSD) is a design (V, B) in which the blocks have some fine structures, or in other words, the blocks satisfy some additional combinatorial conditions. The block forms of a multi-structured design can be described as superblocks containing sub-blocks. Let B = {B1 , B2 , . . . , Bb }. The block Bi has the form Bi = {Ci1 , Ci2 , . . . , Cin }, where Ci j ⊆ Bi , which satisfies one of the following conditions for any i: 1. {Ci1 , C i2 , . . . , Cini } is a partition of Bi ; 2. Bi = 1≤ j≤ni Ci j , where the sub-blocks Ci j are not necessary mutually disjoint.

Finite geometries are typical examples of the case 2. However, we don’t consider the case in this paper. Finite geometries are typical examples of MSDs of case 2. However, we will not consider this case in this paper. ˜ i = {Ci1 , Ci2 , . . . , If Bi is considered as a set of disjoint n i sub-blocks, that is, B ˜ i is called unordered. If Bi is considered as an ordered set Cini }, then the super-block B  i = (Ci1 , Ci2 , . . . , Cin ), then of disjoint n sub-blocks, some of which can be ∅, that is, B  the super-block Bi is called ordered. Multi-structured designs arise in many situations with various applications. In what follows, we describe some typical examples of their applications and related constructions.

2. Experiments Based on Block Designs Consider a multi-structured design (V, B), where B is a collection of super-blocks B = ˜ i = {Ci1 , Ci2 , . . . , Cin }, with {Ci1 , Ci2 , . . . , Cin } being {B1 , B2 , . . . , Bb } such that B i i a partition of Bi for each 1 ≤ i ≤ b. Let C = {Ci j | 1 ≤ i ≤ b, 1 ≤ j ≤ n i }. The super-design (V, B) satisfies the following conditions: 1. the super-block size is a constant k (regular); 2. every point of V appears in the same number r of super-blocks (singleton balance); 3. every pair of distinct points of V appears in the same number λ of super-blocks (pair balance), that is, (V, B) is a balanced incomplete block design. Furthermore, the sub-design (V, C) satisfies the following conditions: ′

1. the sub-block size is a constant k (regular); ′ 2. every point of V appears in the same number r of sub-blocks (singleton balance); ′ 3. every pair of distinct points of V appears in the same number λ of sub-blocks (pair balance),

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′

that is, (V, C) is a B(k , λ ; v). Such an MSD is called a nested design. Historically, different notions of nested designs were introduced by Preece [66] and Federer [26], which were finally unified by Kageyama and Miao [50]. Example 2.1 Let V = {0, 1, 2, 3, 4}, and B = {Bi | 1 ≤ i ≤ 5}, C = {Ci j | 1 ≤ i ≤ 5, 1 ≤ j ≤ 2}, where ˜ 1 = {C11 = {0, 1}, C12 = {2, 4}}, B

B˜ 2 = {C21 = {1, 2}, C22 = {3, 0}},

˜ 3 = {C31 = {2, 3}, C32 = {4, 1}}, B

˜ 4 = {C41 = {3, 4}, C42 = {0, 2}}, B

˜ 5 = {C51 = {4, 0}, C52 = {1, 3}}. B Then (V, B, C) is a multi-structured design B(4, 3; 5) with a sub-desgin B(2, 1; 5). MSDs can do more than conventional designs do. Let’s consider the conventional agricultural experiment with 7 wheat varieties and 7 fertilizers. The following is the standard block experiment, where the rows i are labeled with wheat varieties and the columns j are labeled with fertilizers. × ×

× ×

×

× × × ×

× ×

×

× × ×

× ×

×

×

×

×

The linear model for this experiment is yi j = µ + αi + β j + ǫi j , 1 ≤ i, j ≤ 7 where yi j is the variable observed on the unit with wheat variety i and fertilizer j in the experiment, µ is the central effect, αi and β j are the variety and fertilizer effects, 7  respectively, with i=1 αi = 7j=1 β j = 0, and ǫi j ’s are uncorrelated random variables for technical errors. The number of total combinations of i and j is 49, but we only need to do 21 experiments which are crossed in the table above. The merits of this experiment include (1) less experiments, (2) all effects are estimable easily, (3) good precision of estimations, and (4) equal precisions of estimations. We next consider an experiment with 5 wheat varieties, 10 fertilizers with 5 kinds each of 2 types, say, organic which is denoted ◦, and chemical which is denoted ×, using the multi-structured design in Example 2.1. × × ◦ ◦

◦ ◦ × ×

◦ × × ◦

× ◦ ◦ ×

◦ × × ◦

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The linear model for this experiment is yi jk = µ + αi + β j + γk + ǫi jk , where the sub-block effects (organic or chemical type) γk , k = 0, 1, further satisfy the condition that γ0 + γ1 = 0. 3. Orthogonal Multi-Structured Designs The sub-blocks in the multi-structured designs introduced in Section 1, like the conventional balanced incomplete block designs, are one-dimensional. In fact, the points of each block can also be arranged into a two-dimensional array. A row-column design is a pair (V, B), where V is a set of v points, B = {B1 , B2 , . . . , Bb } is a collections of m × n arrays, called blocks, with entries from V , where ⎞ ⎛ x11 x12 . . . x1m ⎜ x21 x22 . . . x2m ⎟ ⎟ ⎜ Bj = ⎜ . . .. ⎟ . . ⎝ . . . ⎠ xn1 xn2 . . . xnm

such that 1. any point of V cannot appear more than once in any arrays; 2. every point of V appears in the same number r of arrays (singleton balance); 3. every pair of distinct points of V appears in the same number λ of arrays (pair balance); 4. every pair of distinct points of V appears in the same number λ R of rows in all arrays (row pair balance); 5. every pair of distinct points of V appears in the same number λC of columns in all arrays (column pair balance). The linear model for experiments based on row-column designs is established by Srivastava [77] and Singh and Dey [75] as follows: yi jkl = µ + αi + β j + γk + δl + ǫi jkl , where αi is the variety effect,  β j is the  block effect,  row (type) effect, δl is  γk is the the column (kind) effect, with i αi = γ = β = l δl = 0, and ǫi jkl ’s are k k j j uncorrelated random variables for technical errors. In other words, if (V, B) is a row-column design, then it is equivalent to an MSD with two sub-designs satisfying the following conditions: ˜ (R) = {C j1 , C j2 , . . . , C jn }, C jk ⊆ B j , and Each block B j has two kinds of sub-blocks B j (C) = {D j1 , D j2 , . . . , D jm }, D jl ⊆ B j , B˜ j

C = {C jk | 1 ≤ j ≤ b, 1 ≤ k ≤ n}, D = {D jl | 1 ≤ j ≤ b, 1 ≤ l ≤ m}, such that

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1. the super-design (V, B) possesses regular, singleton balance and pair balance properties, 2. the sub-designs (V, C) and (V, D) posses regular and pair balance properties; ˜ (R) and B ˜ (C) are orthogonal for any 1 ≤ j ≤ b, that is, |C jk ∩ D jl | = 1 for 3. B j j 1 ≤ k ≤ n, 1 ≤ l ≤ m. It can be easily proved that each of the two sub-designs also possesses singleton balance property. Example 3.1 The following is an MSD with two orthogonal sub-designs, where V = {0, 1, 2, 3, 4, 5, 6, 7, ∞}, block size is 3 × 3, r = 4, λ = 4, λ R = 1, λC = 1. The blocks are described below: ∞ 1 5

0 2 3

4 7 6

∞ 3 7

2 4 5

6 1 0

∞ 0 4

1 2 7

5 3 6

∞ 2 6

3 4 1

7 5 0

4. Authentication Codes Multi-structured designs also have applications in many disciplines such as cryptology. Authentication codes were invented by Gilbert et. al. [42] for protecting the integrity of information, which involve three active parties: Alice, Bob, and Oscar. Alice and Bob want to communicate over an insecure channel. Oscar, the opponent, has the ability to introduce his own messages into the channel (the impersonation attack) and/or to modify existing messages (the substitution attack). A game-theoretic model for authentication codes was developed by Simmons [72]. In this model, Alice and Bob share a common encoding rule (or key) e. The key e is chosen from a key space E according to some specified probability distribution. Given a source state (or plaintext) s from some source state space S, when Alice wants to communicate s to Bob, she computes a message m = (s, e(s)) ∈ M, where M is the message space and e(s) ∈ A is the authenticator of s, and then sends m ∈ M to Bob over the channel. Bob accepts or rejects the transmitted message m = (s, a) ∈ M based on the key e ∈ E which Bob shared with Alice. If a = e(s), then Bob is able to detect that an attack has taken place. The strength of an authentication code is measured by the deception probabilities P0 and P1 , which represent the probability that Oscar can deceive Bob by impersonation and substitution, respectively. In computing P0 and P1 , it is assumed that Oscar is using an optimal strategy. When Alice and Bob use an authentication code, they want P0 and P1 , as well as |E|, to be small, Example 4.1 The following is an example of authentication code with S = {s0 , s1 , s2 }, E = {e0 , e1 , . . . , e8 }, and A = {0, 1, 2}. The key ei is chosen from E at random. It is easily seen that this authentication code has the deception probabilities P0 = P1 = 1/3 and |E| = 9. In fact, P0 , P1 and |E| are the smallest ones for the case |A| = 3.

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e0 e1 e2 e3 e4 e5 e6 e7 e8

s0 0 0 0 1 1 1 2 2 2

s1 0 1 2 0 1 2 0 1 2

331

s2 0 2 1 1 0 2 2 1 0

It is possible that more than one authenticator can be used to authenticate a particular source state s ∈ S; this is called splitting, an important concept in the context of an authentication code with arbitration. In this case, a message m ∈ M is computed as m = (s, e(s, r )), where r is some random number chosen from a specified finite set R. If we define e(s) = {a ∈ A | a = e(s, r ) for some r ∈ R}, then splitting means that |e(s)| > 1 for some e ∈ E and s ∈ S. It is also required that for any e ∈ E, e(s) ∩ e(s ′ ) = ∅ if s = s ′ . Obviously, Bob accepts m = (s, a) ∈ M if a ∈ e(s). A splitting authentication code is called c-splitting if |e(s)| = c for any e ∈ E and any s ∈ S. Theorem 4.2 [64] For any c-splitting authentication code, P0 ≥ c|S|/|M|,

P1 ≥ c(|S| − 1)/(|M| − 1).

If in fact the above equalities are satisfied, then |E| ≥ |M|(|M| − 1)/(c2 |S|(|S| − 1)). A c-splitting authentication code is optimal if it satisfies all the equalities in Theorem 4.2. An optimal c-splitting authentication code has been shown to be closely related to a multi-structured design called c-splitting balanced incomplete block design. In this paper, however, we will call it a multi-structured design with external balance. The following is its formal definition. Let v, u and c be positive integers such that v ≥ uc. Let V be a v-set of points, B = {B1 , B2 , . . . , Bb } be a collection of super-blocks with entries from V such that  i = (Ci1 , Ci2 , . . . , Ciu ), |Ci j | = c. The each super-block has u sub-blocks of size c, B pair (V, B) is said to be a multi-structured design with external balance if it satisfies the following conditions: (1) every point occurs at most once in each super-block; (2) for every pair of distinct points x, y ∈ V , there is exactly one super-block which contains x, y in distinct sub-blocks (external balance condition).

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Example 4.3 The following is a multi-structured design with external balance with v = 25, u = 3 and c = 2 taken from [39], where the point set is Z25 . The collection of blocks is obtained by developing the elements of Z25 in the following given block +1 modulo 25:

({0, 1}, {2, 4}, {12, 20}).

Ogata et al. [64] showed the following relations between splitting authentication codes and multi-structured designs with external balance. An authentication matrix of a c-splitting authentication code is a matrix with the rows indexed by the keys e ∈ E, the columns indexed by the source states s ∈ S, and entry (e, s) given by e(s) ⊆ A.

Theorem 4.4 [64] If there exists an optimal c-splitting authentication code, then the rows of its authentication matrix form the blocks of an MSD with external balance. The resulting MSD has |M| points, and each of its super-blocks contains |S| sub-blocks of size c. Conversely, starting from an MSD with external balance, (V, B), with parameters v, u and c, we can put A = V , S = {s0 , s1 , . . . , su−1 }, and for each super-block

(C1 , C2 , . . . , Cu ),

we define an encoding rule e ∈ E such that e(s0 ) = C1 , e(s1 ) = C2 , . . . , e(su−1 ) = Cu . Then we obtain the following result.

Theorem 4.5 [64] If there exists an MSD with external balance with parameters v, u and c, then there exists an optimal c-splitting authentication code such that (1) |A| = v, |S| = u; (2) each source state occurs with equal probability. Example 4.6 The following is an example of optimal 2-splitting authentication code with S = {s0 , s1 , s2 }, E = {e0 , e1 , . . . , e24 }, and A = {0, 1, . . . , 24}, constructed from the MSD in Example 4.3.

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e0 e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15 e16 e17 e18 e19 e20 e21 e22 e23 e24

s0 0, 1 1, 2 2, 3 3, 4 4, 5 5, 6 6, 7 7, 8 8, 9 9, 10 10, 11 11, 12 12, 13 13, 14 14, 15 15, 16 16, 17 17, 18 18, 19 19, 20 20, 21 21, 22 22, 23 23, 24 24, 0

s1 2, 4 3, 5 4, 6 5, 7 6, 8 7, 9 8, 10 9, 11 10, 12 11, 13 12, 14 13, 15 14, 16 15, 17 16, 18 17, 19 18, 20 19, 21 20, 22 21, 23 22, 24 23, 0 24, 1 0, 2 1, 3

333

s2 12, 20 13, 21 14, 22 15, 23 16, 24 17, 0 18, 1 19, 2 20, 3 21, 4 22, 5 23, 6 24, 7 0, 8 1, 9 2, 10 3, 11 4, 12 5, 13 6, 14 7, 15 8, 16 9, 17 10, 18 11, 19

5. Orthogonal and Balanced Arrays Now we consider a multi-structured design (V, B), where |V | = v and B is a collec i = (Ci1 , Ci2 , . . . , Cin ), tion of ordered super-blocks B = {B1 , B2 , . . . , Bb } such that B where some of Ci j , 1 ≤ j ≤ n, can be ∅. Let C j = {Ci j | 1 ≤ i ≤ b}, 1 ≤ j ≤ n. The (n + 2)-tuple (V, B, C1 , . . . , Cn ) is called an MSD with mutually balanced sub-designs provided that (V, B, C1 , . . . , Cn ) satisfies the following conditions: 1. (V, B) has the singleton balance property (r ), 2. (V, B) has the pair balance property (λ), that is, (V, B) is an (r, λ)-design; for any 1 ≤ j ≤ n, 3. (V, C j ) has the singleton balance property (r j ), 4. (V, C j ) has the pair balance property (λ j ), that is, each (V, C j ) is an (r j , λ j )-design; furthermore, 5. every pair {x, y} of distinct points of V appears in the same number µi j of superblocks such that x is in the ith sub-block and y is in the jth sub-block (called external balance condition).

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Example 5.1 The following is an MSD with mutually balanced sub-designs having parameters r = 10, λ = 4, r1 = r2 = 5, λ1 = λ2 = λ12 = 1:  1 = ({1, 3}, {2}), B  2 = ({1, 2, 5}, ∅), B  3 = ({6}, {1, 3}), B  4 = ({5}, {1, 6}), B  5 = (∅, {2, 3, 4}), B  6 = ({2, 4}, {5}), B  B7 = ({3, 4}, {6}),  8 = ({6}, {4, 5}), B  9 = ({2}, {1, 4}), B  10 = ({4}, {1, 2}), B

 11 = ({1}, {3, 5}), B  12 = ({3}, {1, 5}), B  13 = ({1, 4, 6}, ∅), B  B14 = ({1}, {4, 6}),  15 = ({2, 3, 6}, ∅), B  16 = ({2}, {3, 6}), B  B17 = ({5, 6}, {2}),  18 = (∅, {2, 5, 6}), B  19 = ({3, 5}, {4}), B  20 = ({4, 5}, {3}). B

 i = (Ci1 , Ci2 , . . . , Cin ), we can define an (n + 1)From any ordered super-block B ary vector of length v, xi = (x0 , x1 , . . . , xv−1 ), where

j if t ∈ Ci j , xt = 0 otherwise. In this way, we can obtain a 20 × 6 array with entries from {0, 1, 2}. The following is the transpose of the array: ⎞ 11220000221211000000 ⎜2 1 0 0 2 1 0 0 1 2 0 0 0 0 1 1 2 2 0 0⎟ ⎟ ⎜ ⎜1 0 2 0 2 0 1 0 0 0 2 1 0 0 1 2 0 0 1 2⎟ ⎟ ⎜ ⎜0 0 0 0 2 1 1 2 2 1 0 0 1 2 0 0 0 0 2 1⎟ ⎟ ⎜ ⎝0 1 0 1 0 2 0 2 0 0 2 2 0 0 0 0 1 2 1 1⎠ 00120021000012121200 ⎛

The above array is in fact an example of balanced arrays. Let S = {0, 1, . . . , s − 1}. A balanced array BA(m, n, s) over S is an n × m array A with entries from S which satisfies the following two conditions: 1. in any two columns of A, any pair (x, y) ∈ S2 occurs exactly µx y times, and 2. for any x, y ∈ S, µx y = µ yx . If µx y = µ for any x, y ∈ S, then the balanced array is an orthogonal array OA(m, n, s). The notion of a balanced array was introduced by Chakravarti [10] as a generalization of that of an orthogonal array and used as a substitute for an orthogonal array in statistics. Example 5.2 The above array in Example 5.1 is the transpose of a BA(6, 20, 3) defined over S = {0, 1, 2} with µ00 = 4, µ01 = µ02 = 3, µ11 = µ12 = µ22 = 1. Lemma 5.3 [53] Let V, B, C j , 1 ≤ j ≤ n, be defined as above, and define C0 = {V \ Bi | 1 ≤ i ≤ b}. If (V, B) is an (r, λ)-design and (V, C j ) is an (r j , λ j )-design for any 1 ≤ j ≤ n, then (V, C0 ) also an (r0 , λ0 )-design with r0 = b − r and λ0 = b − 2r + λ. Furthermore, if condition 5 (external balance) is satisfied for all 1 ≤ i, j ≤ n, then it is also satisfied for all 0 ≤ i, j ≤ n, with µ0i = µi0 = ri − nj=1 µi j .

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Theorem 5.4 [53] There exists an ordered multi-structured design which satisfies conditions 1–5 above if and only if there exists a b × v balanced array with entries from S = {0, 1, . . . , n} and parameters µx y , 0 ≤ x, y ≤ n.

6. Cyclic Multi-Structured Blocks and Sequences The case where V = Zv , where Zv is the residue ring of integers modulo v, is of special interest in modern communications. In what follows, we focus on cyclic multi-structured designs and their related sequences used in communications. Let (V, B) be a design, and σ be a permutation on V . For any block B = {b1 , . . . , bk } ∈ B, define Bσ = {b1σ , . . . , bkσ }. If Bσ = {Bσ | B ∈ B} = B, then σ is an automorphism of the design (V, B). The set of all such permutations forms a group under composition called the full automorphism group of the design. Any of its subgroups is an automorphism group of the design. A design admitting a cyclic automorphism group is a cyclic design. For a cyclic design (V, B), the point set V can be identified with Zv , The cyclic automorphism then is just the bijection σ : i −→ i + 1 (mod v). Cyclic designs (Zv , B) having only full block orbits are of particular interest. In this section, when we say a cyclic design, we always mean a cyclic design in which each of its block orbits under the automorphism σ contains exactly v distinct blocks. Example 6.1 A cyclic B(3, 1; 7) on Z7 . B0 = {0, 1, 3}, B4 = {4, 5, 0},

B1 = {1, 2, 4}, B5 = {5, 6, 1},

B2 = {2, 3, 5}, B6 = {6, 0, 2}.

B3 = {3, 4, 6},

Consider an (n + 1)-ary sequence X = (x0 , x1 , . . . , xv−1 ) of length v based on an alphabet Q = {0, 1, 2, . . . , n}. The cyclic shift function ϕ is defined by ϕ(X ) = (xv−1 , x0 , x1 , . . . , xv−2 ).  = (C1 , C2 , . . . , Cn ), We can construct a corresponding ordered multi-structured block B where Ci ’s are the supports of X , that is, Ci = supp X (i) = {j | x j = i, 0 ≤ j < v}, 1 ≤ i ≤ n. We can also define C0 = Zv \

n 

Ci .

i=1

 = (C1 , C2 , . . . , Cn ), we can Conversely, from an ordered multi-structured block B also construct a corresponding sequence based on Q = {0, 1, 2, . . . , n}. Example 6.2 The following multi-structured blocks  = (C1 = {1, 2, 5}, B  σ = (C σ = {2, 3, 6}, B 1

C2 = {3, 6, 8}), C2σ = {4, 7, 0})

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on Z9 correspond to the sequences X = (0, 1, 1, 2, 0, 1, 2, 0, 2), ϕ(X ) = (2, 0, 1, 1, 2, 0, 1, 2, 0). In the subsequent sections, we will explain how people solve the problems in sequences by transforming them into the problems in cyclic multi-structured designs.

7. Differences and Hamming Correlations Consider any two (n + 1)-ary sequences of length v, X = (x 0 , x1 , . . . , xv−1 ) and Y = (y0 , y1 , . . . , yv−1 ), on an alphabet Q = {0, 1, 2, . . . , n}. The Hamming correlation of X and Y is defined to be n (z) = max { H X,Y (t)},

if X = Y,

n (z) H X,Y = max { H X,Y (t)},

if X = Y,

H X,Y

0≤t g(λ) is uniquely embeddable in a symmetric (v + k + λ, k + λ, λ) design. Neumaier [94] later reduced the bound g(λ) considerably. Theorem 8.4. A quasi-residual 2-(v, k, λ) design is embeddable if either λ = 3, and k > 76, or λ = 3, and k > 12 (λ2 − 1)(λ3 − λ2 − λ + 2). An improvement of the above result is the following theorem due to Metsch [91]. Theorem 8.5. A quasi-residual 2-(v, k, λ) design is embeddable provided that k>



8 √ λ + λ + 5 (λ − 1) λ2 . 3

Between 1944 and 1978, many examples of non-embeddable designs were constructed. Most of these examples are either of Bhattacharya type or non-embeddable because the corresponding symmetric design does not exist. Based on these results, researchers tended to believe that the existence of non-embeddable designs is exceptional. However, in recent years other techniques were used and infinite families of non-

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embeddable designs were constructed. Many of these families correspond to known families of symmetric designs. These new results give some evidence to believe that, if there is a residual design, then there is a non-embeddable quasi-residual design with same parameters except for some few cases. 9. Non-embeddability conditions There could be several reasons for a quasi-residual design to be non-embeddable. We refer to such reasons as non–embeddability conditions. A trivial type of non–embeddability condition for a quasi-residual design is where the associated symmetric design does not exist. The non-existence of the corresponding symmetric designs is shown using the B-RC Theorem. In [122], van Lint gave a family that satisfies this type of non-embeddability condition. Let n ≥ 3 be an integer and let Dn be an (n − 1)-fold multiple of a symmetric (n, n − 1, n − 2) design. Then, Dn is a quasi-residual design with parameters (n, n(n − 1), (n − 1)2 , n − 1, (n − 1)(n − 2)). If Dn is embeddable in a symmetric design Sn , then the complement of Sn has parameters (n2 − n + 1, n, 1) which are those of a projective plane of order n − 1. This leads to the following. Corollary 9.1. Let n ≥ 3 be an integer. If there is no projective plane of order n − 1, then there exists a non-embeddable quasi-residual design with parameters (n, n(n − 1), (n − 1)2 , n − 1, (n − 1)(n − 2)). One of the most common methods that are used to establish the non-embeddability of a quasi-residual design is based on the intersections of the blocks. The main idea of this type is to show that the design contains a collection of blocks with a specified pairwise intersection sizes that prevents the design from being embedded. For example, a Bhattacharya type design is non-embeddable because it has a pair of blocks intersecting in more than λ points. Non-embeddability conditions that are based on block intersections have been widely used (see [47,54,113,118,122,128,129]). We give the following example from [128] as an illustration on this technique. Example 9.1. Let X = {1, 2, . . . , 25} and let B = {B1 , . . . , B40 } be the set of the following 40 subsets of X: B21 = {1, 3, 4, 6, 8, 9, 21, 22, 25, 11} B1 = {1, 3, 4, 7, 10, 13, 14, 18, 19, 21} B2 = {2, 4, 5, 8, 6, 14, 15, 19, 20, 22} B22 = {2, 4, 5, 7, 9, 10, 22, 23, 21, 12} B3 = {3, 5, 1, 9, 7, 15, 11, 20, 16, 23} B23 = {3, 5, 1, 8, 10, 6, 23, 24, 22, 13} B4 = {4, 1, 2, 10, 8, 11, 16, 17, 24} B24 = {4, 1, 2, 9, 6, 7, 24, 25, 23, 14} B5 = {5, 2, 3, 6, 9, 12, 13, 17, 18, 25} B25 = {5, 2, 3, 10, 7, 8, 25, 21, 24, 15} B6 = {6, 7, 10, 12, 15, 18, 19, 22, 25, 1} B26 = {11, 13, 14, 3, 4, 6, 7, 10, 17, 20} B7 = {7, 8, 6, 13, 11, 19, 20, 23, 21, 2} B27 = {12, 14, 15, 4, 5, 7, 6, 8, 6, 18, 16} B8 = {8, 9, 7, 14, 12, 20, 16, 24, 22, 3} B28 = {13, 15, 11, 5, 1, 8, 9, 7, 19, 17} B9 = {9, 10, 8, 15, 13, 16, 17, 25, 23, 4} B29 = {14, 11, 12, 1, 2, 9, 10, 8, 20, 18} B10 = {10, 6, 9, 11, 14, 17, 18, 21, 24, 5} B30 = {15, 12, 13, 2, 3, 10, 6, 9, 16, 19} B11 = {21, 23, 24, 3, 4, 12, 15, 17, 20, 6} B31 = {11, 12, 15, 7, 10, 21, 22, 25, 17, 20}

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B12 = {22, 24, 25, 4, 5, 13, 11, 18, 16, 7} B32 = {12, 13, 11, 8, 6, 22, 23, 21, 18, 16} B13 = {23, 25, 21, 5, 1, 14, 12, 19, 17, 8} B33 = {13, 14, 12, 9, 7, 23, 24, 22, 19, 17} B14 = {24, 21, 22, 1, 2, 15, 13, 20, 18, 9} B34 = {14, 15, 13, 10, 8, 24, 25, 23, 20, 18} B15 = {25, 22, 23, 2, 3, 11, 14, 16, 19, 10} B35 = {15, 11, 14, 6, 9, 25, 21, 24, 16, 19} B16 = {17, 18, 19, 20, 3, 4, 8, 9, 22, 25} B36 = {11, 12, 15, 23, 24, 1, 3, 4, 18, 19} B17 = {18, 19, 20, 16, 4, 5, 9, 10, 23, 21} B37 = {12, 13, 11, 24, 25, 2, 4, 5, 19, 20} B18 = {19, 20, 16, 17, 5, 1, 10, 6, 24, 22} B38 = {13, 14, 12, 25, 21, 3, 5, 1, 20, 16} B19 = {20, 16, 17, 18, 1, 2, 6, 7, 25, 23} B39 = {14, 15, 13, 21, 22, 4, 1, 2, 16, 17} B20 = {16, 17, 18, 19, 2, 3, 7, 8, 21, 24} B40 = {15, 11, 14, 22, 23, 5, 2, 3, 17, 18} Then, D = (X, B ) is a 2 − (25, 10, 6) design. Assume that D is embeddable in a symmetric (41, 16, 6) design S. Now, consider the blocks B1 , B26 , B38 , and B39 . Let Ci be the block of S that contains Bi , and let {26, 27, . . . , 41} be the set of new points. Now, we have |B1 ∩ B26 | = 6, |B1 ∩ B38 | = 5, and |B26 ∩ B38 | = 4. So, without loss of generality, we may assume that C1 = B1 ∪ {26, 27, 28, 29, 30, 31}, C26 = B26 ∪ {32, 33, 34, 35, 36, 37}, C38 = B38 ∪ {26, 32, 33, 38, 39, 40}. Also, we have |B1 ∩ B39 | = 5, |B26 ∩ B39 | = 4, and |B38 ∩ B39 | = 5. Then, C39 can have at most 4 points from the set {26, 27, . . . , 40}. But we need 6 points for C39 , a contradiction. Hence D is non-embeddable. Ionin and Mackenzie-Fleming [54] gave the following general definition of nonembeddability conditions that depends on the intersection of the blocks. Definition 9.1. Let Pm be the set of all polynomials m

m

f = a + a0 x0 + ∑ ∑ ai j xi j i=1 j=1

in 1+m2 variables x0 , xi j with integer coefficients a0 , ai j and the free term a equal to 0 or 1. Let F be a subset of Pm . We will say that a 2-(v, k, λ) design D satisfies the inequality F > 0 (resp. F ≥ 0), if D has m distinct blocks B1 , B2 , · · · , Bm such that, if x0 = λ and  xi j = |Bi B j | for i, j = 1, 2, · · · , m, then the value of each polynomial f ∈ F is positive (resp. nonnegative). The set F is called an m block non-embeddability condition, if every quasiresidual design which satisfies the inequality F > 0 is non-embeddable. Such m block non-embeddability conditions are very useful in constructing infinite families of nonembeddable quasi-residual designs. Usually, an infinite family is constructed recursively with the property that, if the initial design satisfies an m block non-embeddability condition, then so do all designs in the family. Bhattacharya type designs satisfy a two block non-embeddability condition as can be seen by taking F = {−x0 + x12 }. The following is a three block non-embeddability condition. The proof is taken from [58].

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Theorem 9.1. The set F = {−x0 − x11 + x12 + x13 + x23 } is a three block nonembeddability condition. Proof. Let D be a quasi-residual (v, b, r, k, λ) design and suppose that B1 , B2 , and B3 be distinct blocks of D that satisfies the inequality F > 0. Then |B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 | > r. Suppose that D is embeddable in a symmetric (v + r, r, λ) design S. For i = 1, 2, 3 let Ai be the block of S that contains Bi and Ci = Ai \ Bi . Then for i = j, |Ci ∩C j | = λ − |Bi ∩ B j |. Therefore, r ≥ |C1 ∪C2 ∪C3 | ≥ |C1 | + |C2 | + |C3 | − (|C1 ∩C2 | + |C1 ∩C3 | + |C2 ∩C3 |) = 3(r − k) − 3λ + (|B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 |) = |B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 | > r, a contradiction. Hence, D is non-embeddable. Next, we give a 2 − (12, 6, 5) design constructed by Mackenzie-Fleming [85] that satisfies this three block non-embeddability condition. Example 9.2. Let X = {1, 2, . . . , 12} and let B = {B1 , . . . , B22 } be the set of the following 22 subsets of X: B2 = {1, 2, 6, 8, 10, 11} B3 = {2, 3, 6, 8, 11, 12} B1 = {3, 5, 6, 8, 10, 11} B4 = {1, 2, 3, 7, 8, 9} B5 = {4, 5, 6, 7, 8, 12} B6 = {1, 2, 4, 7, 8, 10} B7 = {1, 2, 5, 7, 11, 12} B8 = {3, 4, 6, 7, 8, 9} B9 = {3, 4, 5, 7, 10, 11} B10 = {1, 3, 4, 9, 10, 11} B11 = {2, 5, 6, 7, 9, 10} B12 = {1, 3, 5, 8, 9, 12} B13 = {2, 4, 6, 9, 10, 12} B14 = {1, 3, 6, 7, 10, 12} B15 = {2, 4, 5, 8, 9, 11} B16 = {1, 4, 5, 8, 10, 12} B17 = {1, 4, 6, 9, 11, 12} B18 = {2, 3, 5, 9, 10, 12} B19 = {1, 5, 6, 7, 9, 11} B20 = {2, 3, 4, 7, 11, 12} B21 = {1, 2, 3, 4, 5, 6} B22 = {7, 8, 9, 10, 11, 12} Then, D = (X, B ) is a 2 − (12, 6, 5) design. Blocks B1 , B2 , and B3 satisfy the inequality in Theorem 9.1. Therefore, D is non-embeddable. Next, we state another three block non-embeddability condition from [4], which is used to produce non-embeddable quasi-residual designs related to Menon designs. Theorem 9.2. The set F = {x12 − x13 − x23 } is a three block non-embeddability condition. Next, we give a non-embeddable quasi-residual 2 − (21, 9, 6) design from [4], which is non-embeddable due to Theorem 9.2. This design corresponds to a Menon design. Example 9.3. Let X = {1, 2, . . . , 21} and let B = {B1 , . . . , B35 } be the set of the following 35 subsets of X: B2 = {1, 2, 5, 9, 11, 13, 14, 16, 18} B1 = {1, 2, 3, 4, 5, 6, 11, 14, 15} B3 = {3, 7, 8, 11, 12, 16, 17, 19, 20} B4 = {1, 2, 3, 4, 8, 13, 19, 20, 21}

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B5 = {1, 2, 4, 7, 10, 12, 13, 15, 17} B6 = {1, 2, 3, 9, 10, 12, 18, 19, 20} B7 = {1, 3, 6, 7, 11, 12, 13, 18, 21} B8 = {1, 4, 6, 7, 8, 9, 11, 13, 19} B9 = {1, 6, 7, 9, 10, 14, 17, 20, 21} B10 = {1, 4, 5, 6, 8, 16, 17, 18, 20} B11 = {1, 5, 6, 12, 15, 16, 19, 20, 21} B12 = {1, 8, 9, 12, 13, 14, 15, 18, 20} B13 = {1, 4, 9, 10, 14, 16, 17, 19, 21} B14 = {1, 3, 5, 7, 8, 15, 17, 18, 21} B15 = {1, 3, 7, 10, 11, 14, 15, 16, 19} B16 = {2, 3, 6, 8, 10, 11, 14, 20, 21} B17 = {2, 4, 6, 7, 8, 14, 16, 18, 19} B18 = {2, 3, 6, 9, 13, 15, 16, 17, 21} B19 = {2, 5, 6, 7, 8, 9, 10, 15, 20} B20 = {2, 6, 7, 12, 14, 15, 17, 18, 19} B21 = {2, 5, 7, 9, 11, 13, 17, 19, 20} B22 = {2, 4, 11, 12, 14, 17, 18, 20, 21} B23 = {2, 8, 10, 13, 15, 16, 18, 19, 21} B24 = {2, 3, 4, 5, 7, 9, 12, 16, 21} B25 = {5, 6, 10, 11, 13, 17, 18, 19, 21} B26 = {4, 6, 10, 11, 12, 13, 15, 16, 20} B27 = {3, 6, 8, 9, 12, 13, 14, 16, 17} B28 = {3, 4, 5, 6, 9, 10, 12, 18, 19} B29 = {1, 2, 5, 8, 10, 11, 12, 16, 17} B30 = {3, 5, 7, 10, 13, 14, 16, 18, 20} B31 = {3, 4, 8, 9, 10, 11, 15, 17, 18} B32 = {4, 7, 9, 11, 15, 16, 18, 20, 21} B33 = {4, 5, 7, 8, 10, 12, 13, 14, 21} B34 = {3, 4, 5, 13, 14, 15, 17, 19, 20} B35 = {5, 8, 9, 11, 12, 14, 15, 19, 21} Then, D = (X, B ) is a 2 − (15, 6, 5) design. Blocks B1 , B2 , and B3 satisfy the inequality F > 0, where F = {x12 − x13 − x23 }. Hence, by Theorem 9.2, this design is nonembeddable. Tonchev [113] introduced another very interesting method to determine the nonembeddability of a given quasi-residual or quasi-derived design using codes and graphs. Let Fp be the finite field of prime order p. An [n, k] code C over F p is a k-dimensional subspace of Fnp . The elements of C are called codewords. For a codeword x ∈ C, the weight of x, denoted as wt(x) is defined to be the number of non-zero coordinates in x. The dual code C⊥ of a code C consists of all vectors in Fnp that orthogonal to all vectors in C. Let D be a quasi-residual (v, b, r, k, λ) design with an incidence matrix A, and let p be a prime divisor of λ. Let C p (D) be the code over FP generated by the rows of A, and let V be the set of all (0, 1)-vectors x in the dual code of CP (D) such that wt(x) = r − 1 and wt(x − y) = 2r − 2λ − 1 for every row y of A. Now, define the graph Γ(D) with set of vertices V and two vertices x and y ∈ V are adjacent if and only if wt(x − y) = 2r − 2λ. We now state the main theorem from Tonchev [113]. Theorem 9.3. The quasi-residual (resp. quasi-derived) design D is embeddable if and only if the graph Γ(D) has an r-clique. Proof. The design D is embeddable in a symmetric (v + r, r, λ) design S if and only if S < ¯= has an incidence matrix of the form N = AA′ 0j , where A′ is an incidence matrix for the corresponding quasi-derived design D′ . Therefore, the embeddability of the design D is equivalent to the existence of the matrix A′ . Obviously, the rows of A′ form an r-clique in the graph Γ(D). Conversely, If Γ(D) has an r-clique H, then the vertices of H form an incidence matrix A′ of a quasi-derived design completing the matrix A to a symmetric (v + r, r, λ) design. For if A′ is a matrix whose rows are vertices of an r-clique then each row of A′ has weight r − 1 and the scalar product of each pair of different rows of A′ is equal to λ − 1. It remains only to show that A′ has a constant column sum equal to λ. But, this is implied by the following lemma.

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λv(v−1) Lemma 9.1. Let v, k, and λ, be positive integers such that r = λ(v−1) k−1 and b = k(k−1) are positive integers. Let X be v × b matrix with entries equal to 0 or 1 such that XX T = (r − λ)I + λJ. Then, JX = kJ, i.e. X is an incidence matrix of a (v, b, r, k, λ) design.

Proof. Let ki be the sum of ith column of X. By counting the ones   in the  scalar products of pairs of rows of X in two ways we obtain ∑ ki = vr and ∑ k2i = λ 2v . This implies that

k 2k − 1 0 ≤ ∑(ki − k)2 = v(v − 1)λ 1 − = 0. + k−1 k−1 Hence k1 = . . . = kb =

vr b

= k.

This method was used by Tonchev [113] to check the embeddability of some of the designs in Preece [97], where he obtained the following result: Among the twelve 2 − (14, 7, 6) designs listed in [97] only four designs are non-embeddable and the other eight are uniquely embeddable in symmetric (27, 13, 6)-designs. In addition, Tonchev [113] established the non-embeddability of quasi-residual designs with parameters 2 − (13, 7, 7) and 2 − (11, 6, 6). Recently, Harada and Miyabayashi [41], with a slight modification of technique, established the existence of non-embeddable quasi-derived designs with parameters 2 − (13, 4, 3), 2 − (15, 6, 5), and 2 − (16, 6, 5). We give the following example from [113] as an application of this technique. Let D be the quasi-residual (14, 26, 13, 7, 6) design defined by the incidence matrix ⎤ 10110001001110100011100011 ⎢0 1 0 1 1 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 1 1 0 0 1 1⎥ ⎥ ⎢ ⎢0 0 1 0 1 1 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 0 1 1⎥ ⎥ ⎢ ⎢1 0 0 1 0 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 0 1 1 1 1 1⎥ ⎥ ⎢ ⎢1 1 0 0 1 0 0 0 1 1 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1⎥ ⎥ ⎢ ⎢0 1 1 0 0 1 1 0 0 1 1 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1⎥ ⎥ ⎢ ⎢1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 0 1 1 0 1 0 1 0 0⎥ ⎥ ⎢ A=⎢ ⎥ ⎢0 1 0 1 1 0 0 1 0 1 1 0 0 0 0 1 1 1 1 1 1 0 1 0 0 0⎥ ⎢0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 0 1 0 0⎥ ⎥ ⎢ ⎢1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 1 1 0 1 1 1 0 0 0⎥ ⎥ ⎢ ⎢1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 0 0 1 0 1 1 1 0 0⎥ ⎥ ⎢ ⎢0 1 1 0 0 1 0 1 1 0 0 1 0 1 1 1 0 0 1 0 1 0 1 1 0 0⎥ ⎥ ⎢ ⎣1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 0⎦ 00000011111111111100000001 ⎡

The dual code of C2 (D) is a [26, 12] code which has 1123 vectors of weight 12. Only one of these vectors has scalar product 6 with all rows of A. Therefore, the associated graph Γ(D) has only one vertex, and hence D is non-embeddable. Other non-embeddability conditions were effective in certain situations. Van Lint and Tonchev [124] and Kageyama and Miao [71] considered quasi-derived designs, and obtained a certain inequality type non-embeddability condition. As a generalization of their work, Alraqad [1] derived the following divisibility type non-embeddability condition.

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Theorem 9.4. Let D be a quasi-derived 2 − ((k − 1)v + 1, k, k − 1) design that has a 2–(v, k, k − 1) subdesign D0 . If D is embeddable, then v ≡ 1 (mod k − 1). Remark 9.1. Most of the time, quasi-residual designs with k ≤ 2v are non-embeddable because they satisfy non-embeddability conditions based on block intersections. However, this type of condition is hardly ever satisfied by a quasi-residual design with k > 2v . Usually, B-R-C Theorem, codes and graphs, and inequality or divisibility nonembeddability conditions are more effective in case of designs with k > 2v . 10. Construction techniques Many different techniques and tools have been used to construct quasi-residual designs. Some of these techniques are useful in constructing a design with a specific parameter set. For example, automorphism groups were used by van Trung [128] to construct designs with parameters 2–(25,10,6) and 2–(36,16,12), and by van Lint, Tonchev, and Landgev [125] to construct a 2–(28,10,5) design. See [115] for more on automorphism groups, codes, and graphs. Tonchev [118] used codes and graphs to construct quasi-residual designs. Starting with an incidence matrix A of a quasi-residual 2-(v, k, λ) design, we delete some of the rows of A, then we find all possible ways to fill them back using codes and graphs. This method works effectively for quasi-residual designs with small parameters. Using computer software, Alraqad and M. S. Shrikhande [3,4] were able to construct many small quasi-residual designs, then they found some of the non-embeddable ones by applying an m block non-embeddability condition. Other construction techniques are very effective in obtaining families of nonembeddable quasi-residual designs. Some of these methods rely on resolvable designs and transversal designs. A group divisible design GDD(k, λ, g; ng) is a triple (Y, G , F ), where Y is a set of ng points, G is an n-collection of g–subsets of Y (called groups) which partition Y , and F is a collection of k–subsets of Y (called blocks) such that every block intersects any group in at most one point and every pair of points from different groups occurs in exactly λ blocks. A transversal design T Dλ [k, g] is a GDD(k, λ, g; kg). So, a transversal design is a group divisible design in which every block intersects each group in exactly one point. We simply refer to a T D1 [k, g] as a T D[k, g]. For example, the dual of an affine plane of order q is a T D[q + 1, q]. A design D = (X, B ) is said to be α-resolvable if B can be partitioned into classes, called α-classes, all of the same size, such that every element of X occurs in exactly α blocks from each class. A resolvable design is simply a 1−resolvable design and a 1−class is called a parallel class. We state the next result due to van Trung [129], where resolvable designs are used to construct quasi-residual designs. For an alternative proof, see [58, Theorem 5.3.10] or [88]. Theorem 10.1. If there is a resolvable (v, b, r, k, λ) design R and if there are, not necessarily isomorphic, (v1 , b1 , r1 , k1 , λ1 ) designs D1 , D2 , · · · , Dr , where v1 = kv , then there exists a 2–design D with parameters (v, rb1 , rr1 , kk1 , r1 λ + (r − λ)λ1 ). Moreover, if k1 r−λ , = k r1 − λ1 then, D is quasi-residual.

(15)

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If the designs R and D1 , D2 , · · · , Dr are quasi-residual, then (15) is satisfied. Furthermore, if some Di is of Bhattacharya type, then so is D. This leads to the following result, which was used by van Trung [129] to construct several families of non-embeddable quasi-derived designs. Corollary 10.1. If there is a quasi-residual 2 − (q, k, λ) design E, where q is a prime power, then for every m ≥ 1, there exists a quasi-residual design D with parameters

qm−1 − 1 2 − qm , qm−1 k, (k + λ) + qm−1 λ . q−1 Furthermore, if E is of Bhattacharya type, then so is D. Proof. Choose R to be the resolvable design AGm−1 (m, q) and for i = 1, 2, . . . , r, choose Dr to be the design E. Then, apply Theorem 10.1. The next theorem due to Alraqad [1], uses α-resolvable designs to construct quasiderived designs. Using this result, Alraqad [1] obtained several families of quasi-derived designs that are non-embeddable due to Theorem 9.4. Theorem 10.2. Let v and k be positive integers and let D0 be a quasi–derived 2-(v, k, k − 1) design. Then the following are equivalent: 1. There exists a 2–((k − 1)v + 1, k, k − 1) design D that contains D0 as a subdesign. 2. There exists a (k − 1)–resolvable 2–((k − 2)v + 1, k − 1, k − 1) design R. Furthermore, such a design D has an incidence matrix of the form

N=

! M E1 E2 · · · Ev , O C1 C2 · · · Cv

where M is an incidence matrix of D0 , O is the all zero matrix, C = [C1 C2 · · · Cv ] is an incidence matrix of the (k − 1)-resolvable design R, and for i = 1, · · · , v, Ei is the v × ((k − 2)v + 1) matrix having all entries in the ith row equal to 1 and all other entries equal to 0. Recently, Tonchev [120] used transversal designs and generalized Hadamard matrices to construct an infinite family of non-embeddable quasi-residual designs. Theorem 10.3. Let D1 = (X, B ) be a T Dμ [qμ, q] design and D2 be a 2–(qμ, μ, μ−1 q−1 ) design (μ ≥ 1, q ≥ 2), and f be a one-to-one mapping between the point set of D2 and the collection of classes of D1 . For each block B of D2 , define a new block f (B) ⊆ X of size qμ obtained by replacing each point x ∈ B by the point class f (x). Then we have: μ−1 ) design D. 1. The collection of new blocks B ′ = { f (B) : B ∈ B } is a 2-(q2 μ, qμ, q−1 2. The design D is resolvable if and only if D1 and D2 are both resolvable. 3. The design D is quasi-residual.

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As application of this theorem, Tonchev [120] showed that there are more than 1017 non-isomorphic quasi-residual 2 − (63, 21, 10) designs. He also constructed a family of quasi-residual designs with parameters

7 · 3m−1 − 1 . 2 − 7 · 3m , 7 · 3m−1 , 2 Ionin [47,48] and Ionin and Mackenzie-Fleming [54] introduced a very useful technique of constructing non-embeddable quasi-residual designs by applying balanced generalized weighing matrices and generalized Hadamard matrices. The main idea of this technique is to construct a family of quasi-residual designs recursively from a quasiresidual design. This family is obtained in a certain way that, if the starter design satisfies an m block non-embeddability condition, then so do all members of the family. For more details on this technique we refer to [47], [48], [54], and [58, Chapter 13]. We also include more applications of using GH matrices to construct families of non-embeddable designs in the next two sections. The following theorem of Ionin [48] uses BGW matrices with parameters as in (1) to construct quasi-residual designs. r−1 Theorem 10.4. Let r be an odd prime power and let D be an (r +1, 2r, r, r+1 2 , 2 ) design satisfying an m block non-embeddability condition. Then, for any positive integer n, there exists a quasi-residual design with parameters



(r + 1)(rn − 1) 2r(rn − 1) n (r + 1)rn−1 (r − 1)rn−1 , ,r , , r−1 r−1 2 2



,

(16)

satisfying an m block non-embeddability condition. Proof. Let X be an incidence matrix of D and let M = {X, J − X}. Let S = {1, σ} be a group of order 2 acting on M (with σ(X) = J − X). Then S is a group of symmetries of M . Since r is an odd prime power, Theorem 5.5 implies that, for any positive integer n, there exists a

n+1 −1 n n r , r , r − rn−1 . BGW r−1 over S. Let W be such a matrix, and we assume that W is normalized. By Theorem 6.2, N = W ⊗ X is an incidence matrix of a design E with parameters (16). Let F > 0 be an m block non-embeddability condition satisfied by D and let B1 , B2 , . . . , Bm be the corresponding m blocks of D. without loss of generality, we assume that these blocks correspond to the first m columns of X. let B′1 , B′2 , . . . , B′m be the blocks of E corresponding to the first m columns of N. Since W is normalized, we obtain that, for all i, j ∈ {1, 2, . . . , m}, |B′i ∩ B′j | = rn |Bi ∩ B j |. Therefore, E satisfies the condition F > 0. 11. Quasi-residual Hadamard designs This section deals with quasi-residual and quasi-derived designs with parameters corresponding to Hadamard designs. Such designs have parameters 2 − (2k, k, k − 1) and 2 − (2k − 1, k − 1, k − 2) respectively. In [126], van Lint, van Tilborg, and Wiekema

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showed that all 2 − (2k, k, k − 1) designs with k ≤ 5 are embeddable. Mackenzie-Fleming [85,86] constructed quasi-residual designs with parameters 2 − (12, 6, 5), 2 − (16, 8, 7), and 2−(20, 10, 9), and Alraqad [2] constructed designs with parameters 2−(14, 7, 6) and 2 − (18, 9, 8). All of these designs satisfy the three block non-embeddability condition in Theorem 9.1. Investigating the non-embeddability of quasi-derived 2 − (2k − 1, k − 1, k − 2) designs is equivalent to investigating their complementary designs, which are quasiresidual designs that corresponds to the complement of Hadamard designs. Such designs have parameters 2 − (2k − 1, k, k). Tonchev [113] used codes and graphs to show the nonembeddability of twelve 2 − (11, 6, 6) designs. Alraqad [2] constructed non-embeddable designs with parameters 2 − (2k − 1, k, k), where k ∈ {7, 8, 9, 10}. All of these designs satisfy the three block non-embeddability condition in Theorem 9.1. Alraqad and M.S. Shrikhande [3] used generalized Hadamard matrices to produce some infinite families of non-embeddable quasi-residual designs with parameters 2 − (2k, k, k − 1) and 2 − (2k − 1, k, k). Theorem 11.1. Let D be a quasi-residual 2−(2k, k, k −1) satisfying the three block nonembeddability condition F > 0, where F = {−x0 − x11 + x12 + x13 + x23 }. If there exists a Hadamard matrix of order n, then there exists a quasi-residual 2–(2nk, nk, nk − 1) design E satisfying the same three block non-embeddability condition. Proof. Let X be an incidence matrix of D and let M = {X, J − X}. Let G = {1, α} be a group of order 2 acting on M (with αX = J − X). Let W = [wi j ] be a normalized Hadamard matrix of order n. We can consider W as a normalized GH(G; n/2). Let Y = [yi j ] be an incidence matrix of the Hadamard 2–(n, n/2, n/2 − 1) design. Then the block matrix N = [W ⊗G X Y ⊗ j2k ] is an incidence matrix of a 2–(2nk, nk, nk − 1) design. Now, assume that the blocks B1 , B2 , and B3 of D satisfy the inequality F > 0, where F = {−x0 − x11 + x12 + x13 + x23 }. This implies that |B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 | > 2k − 1. Without loss of generality, we can assume that these blocks correspond to the first three columns of X. Let C1 , C2 , and C3 be the three blocks of E that correspond to the first three columns of N. Then we have |C1 ∩C2 | + |C1 ∩C3 | + |C2 ∩C3 | = n(|B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 |) ≥ n(2k) > 2nk − 1. Therefore, E satisfies the three block non-embeddability condition in Theorem 9.1. Applying Theorem 11.1 with appropriate starter designs gives the following result. Corollary 11.1. Let k ∈ {6, 7, 8, 9, 10}. If there exists a Hadamard matrix of order n, then there is a non-embeddable quasi-residual 2–(2nk, nk, nk − 1) design. Theorem 11.2. Let D be a quasi-residual 2–(2k − 1, k, k) design satisfying the three block non-embeddability condition F > 0, where F = {−x0 − x11 + x12 + x13 + x23 }. If there exists a Hadamard matrix of order n, then there exists a quasi-residual 2–(2nk − 1, nk, nk) design E satisfying the same three block non-embeddability condition.

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Proof. Let X be an incidence matrix of D and let M = {X, J − X}. Let G = {1, α} be a group of order 2 acting on M (with αX = J − X). Let Y be a normalized Hadamard matrix of order n. Replacing every −1 by α in Y produces a GH(G; n/2) matrix H = [hi j ]. Let W = [wi j ] = 12 (J −Y ), and let N be the block matrix ! W12 ⊗ J2k−1×2 H ⊗G X , N= W21 ⊗ j1×4k−2 W22 ⊗ j1×2 where W12 is the matrix obtained from W by deleting the first column, W21 is the matrix obtained from W by deleting the first row, and W22 is the matrix obtained from W by deleting the first column and the first row. Then N is an incidence matrix of a 2–(2nk − 1, nk, nk) design E. The non-embeddability of the design E can be proven as in Theorem 11.1. Corollary 11.2. Let k ∈ {7, 8, 9, 10}. If there exists a Hadamard matrix of order n, then there exists a non-embeddable quasi-residual 2–(2nk − 1, nk, nk)-design. 12. Quasi-residual Menon designs A Menon design of order h2 is a symmetric (4h2 , 2h2 − h, h2 − h)-design. As we pointed out in Theorem 3.2, the existence of such a design is equivalent to the existence of a regular Hadamard matrix of order 4h2 . Quasi-residual and quasi-derived designs that correspond to a Menon design of order h2 have parameters 2 − (2h2 + h, h2 , h2 − h) and 2 − (2h2 − h, h2 − h, h2 − h − 1) respectively. So, designs with such parameters are referred to as quasi-residual and quasi-derived Menon designs of order h2 . It is clear that quasi-residual and quasi-derived Menon designs of orders 1 and 4 are embeddable. In [129], van Trung constructed a non-embeddable 2 − (21, 9, 6) design using automorphism groups. He used a long argument based on block intersections to show the non-embeddability of this design. Alraqad and M.S. Shrikhande [4] gave another 2 − (21, 9, 6) design (Example 9.3) which is non-embeddable due to Theorem 9.2. In [128], van Trung constructed a non-embeddable quasi-residual Menon design of order 16. We describe this design by its incidence matrix. Example 12.1. Let U = (J − P − I)/2 and let V = (J + P − I)/2, where P is a Paley matrix of order 7. Then the following block matrix is an incidence matrix of quasi-residual 2 − (36, 16, 12) design. ⎡

jT jT jT jT 0 0 0 0 ⎢ U U U U U + I U + I U + I U +I ⎢ ⎢ O U +I V +I U +I V U U V ⎢ ⎢U + I O U + I V + I V V U U ⎢ ⎣V +I U +I O U +I U V V U U +I V +I U +I O U U V V

⎤ 0 O ⎥ ⎥ U +I ⎥ ⎥ U +I ⎥ ⎥ U +I ⎦ U +I

This design has three distinct blocks B1 , B2 , B3 such that |B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 | = 30. Hence, it satisfies the inequality F > 0, where F = {−x0 − x11 + x12 + x13 + x23 }.

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Alraqad and M.S. Shrikhande [4] gave a recursive method of constructing quasiresidual and quasi-derived Menon designs using regular Hadamard matrices. Theorem 12.1. Let h and n be positive integers and let H be a regular Hadamard matrix of order 4h2 . Suppose there exist quasi-residual and quasi-derived Menon designs D and E of order n2 . Then there exist quasi-residual and quasi-derived Menon designs D1 and E1 of order (2nh)2 . Proof. Let X and Y be incidence matrices of the designs D and E respectively. By Remark 3.1, the matrix H can be written in the form ! − j2h2 −h H1 . H= j2h2 +h H2 Define the block matrices X1 and Y1 as follows:

X1 =

j2h2 −h ⊗ (J −Y ) H1 ⊗Y H1 ⊗ j2h2 −h j2h2 +h ⊗ X H2 ⊗ X H2 ⊗ 02h2 +h

!

(17)

Y1 =

! j2h2 −h ⊗ (J − X) H1 ⊗ X H1 ⊗ 02h2 +h . j2h2 +h ⊗Y H2 ⊗Y H2 ⊗ j2h2 +h

(18)

Then X1 and Y1 are incidence matrices for the required designs D1 and E1 respectively. The next theorem due to Alraqad and M.S. Shrikhande [4] gives a sufficient condition for the non-embeddability of the quasi-residual and the quasi-derived designs constructed in Theorem 12.1. Let F be a set of polynomials as in Definition 9.1 with free term a equal to 0. Theorem 12.2. Let h and n be positive integers. Let the designs D, E, D1 , and E1 , and the matrix H be as in Theorem 12.1. Furthermore, suppose that D satisfies an inequality F > 0 and the complementary design of E satisfies the inequality F ≥ 0. Then both of the designs D1 and the complementary design of E1 satisfy the inequality F > 0. In order to apply Theorems 12.1 and 12.2, a quasi-residual and a quasi-derived Menon designs of the same order are needed. Next, we give examples of quasi-derived 2 − (15, 6, 5) and 2 − (28, 12, 11) designs. These designs are embeddable, yet they still work as starter designs for Theorems 12.1 and 12.2 to produce infinite families of nonembeddable designs. Example 12.2. Let X = {1, 2, . . . , 15} and let B = {B1 , . . . , B35 } be the set of the following 35 subsets of X: B2 = {2, 4, 6, 8, 11, 14} B3 = {1, 3, 5, 7, 10, 13} B1 = {2, 4, 6, 9, 12, 15} B4 = {1, 3, 5, 8, 11, 14} B5 = {1, 3, 5, 9, 12, 15} B6 = {1, 2, 3, 4, 5, 6} B7 = {2, 4, 6, 7, 10, 13} B8 = {1, 2, 3, 4, 5, 6} B9 = {5, 6, 7, 10, 11, 12} B10 = {3, 6, 8, 12, 13, 14} B11 = {4, 5, 9, 11, 13, 15} B12 = {1, 2, 7, 13, 14, 15}

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B13 = {2, 5, 8, 11, 13, 15} B14 = {1, 6, 9, 12, 13, 14} B15 = {3, 4, 7, 13, 14, 15} B16 = {1, 4, 8, 10, 14, 15} B17 = {2, 3, 9, 10, 14, 15} B18 = {5, 6, 10, 13, 14, 15} B19 = {3, 6, 7, 8, 11, 15} B20 = {4, 5, 7, 9, 12, 14} B21 = {1, 2, 7, 8, 9, 10} B22 = {2, 5, 7, 9, 11, 14} B23 = {1, 6, 7, 8, 12, 15} B24 = {3, 4, 7, 8, 9, 10} B25 = {1, 4, 8, 9, 11, 13} B26 = {2, 3, 8, 9, 12, 13} B27 = {5, 6, 7, 8, 9, 13} B28 = {3, 6, 9, 10, 11, 14} B29 = {4, 5, 8, 10, 12, 15} B30 = {1, 2, 10, 11, 12, 13} B31 = {2, 5, 8, 10, 12, 14} B32 = {1, 6, 9, 10, 11, 15} B33 = {3, 4, 10, 11, 12, 13} B34 = {1, 4, 7, 11, 12, 14} B35 = {2, 3, 7, 11, 12, 15} Then, D = (X, B ) is a 2 − (15, 6, 5) design. Notice that |B1 ∩ B2 | − |B1 ∩ B3 | − |B2 ∩ B3 | = 3. Hence, the complementary design of D satisfies the inequality F ≥ 0, where F = {x12 − x13 − x23 }. Example 12.3. Let U = (J − P − I)/2 and let V = (J + P − I)/2, where P is a Paley matrix of order 7. Then, the following matrix is an incidence matrix of a quasi-derived 2 − (28, 12, 11) design. ⎡

0 0 0 0 jT jT jT jT ⎢ U U U U U + I U + I U + I U +I ⎢ ⎢ O U +I V +I U +I V U U V ⎢ ⎢U + I O U + I V + I V V U U ⎢ ⎣V +I U +I O U +I U V V U U +I V +I U +I O U U V V

⎤ 0 O ⎥ ⎥ U +I ⎥ ⎥ U +I ⎥ ⎥ U +I ⎦ U +I

This design has three distinct blocks B1 , B2 , B3 such that |B1 ∩ B2 | + |B1 ∩ B3 | + |B2 ∩ B3 | = 24. Hence, its complementary design satisfies the inequality F ≥ 0, where F = {−x0 − x11 + x12 + x13 + x23 }. Applying Theorems 12.1 and 12.2 on the designs in Examples 9.3 and 12.2 yields the following result. Corollary 12.1. Let h be a positive integer. If there exists a regular Hadamard matrix of order 4h2 , then there exist non-embeddable quasi-residual and quasi-derived Menon designs with parameters 2–(72h2 + 6h, 36h2 , 36h2 − 6h) and 2–(72h2 − 6h, 36h2 − 6h, 36h2 − 6h − 1) respectively. Applying Theorems 12.1 and 12.2 on the designs in Examples 12.1 and 12.3 yields the following result. Corollary 12.2. Let h be a positive integer. If there exists a regular Hadamard matrix of order 4h2 , then there exist non-embeddable quasi-residual and quasi-derived Menon designs with parameters 2–(128h2 + 8h, 64h2 , 64h2 − 8h) and 2–(128h2 − 8h, 64h2 − 8h, 64h2 − 8h − 1) respectively. Theorem 3.3 and Corollaries 12.1 and 12.2 give the following result. Theorem 12.3. There exist non-embeddable quasi-residual and quasi-derived Menon designs of orders 36h2 and 64h2 , where 1. h = 2 · 3m , m is positive integer; 2. h = 2 · t 2 , t is an odd integer;

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3. h − 1 and h + 1 are both prime powers; 4. h = 4qm, q = 8m2 − 1 is a prime power and there exists a Hadamard matrix of order 4m. 5. h is a positive integer and h or 2h or 3h or 6h is a square. 13. Some remarks and open problems In [122], van Lint includes a table listing possible parameter sets of quasi-residual designs with v ≤ 16. His table indicated that all quasi-residual designs with v ≤ 10 are embeddable. Also all parameters (with v ≤ 16) for which a non-embeddable quasi-residual design exists are settled. The existence of non-embeddable quasi-derived 2 − (v, k, k − 1) designs with v ≤ 16 are determined except for the parameter sets 2 − (12, 3, 2), 2 − (15, 3, 2), and 2 − (16, 5, 4) (See [41, Table 1]). We point out that the case 2 − (16, 3, 2) has been settled in [3]. It is of interest to determine the remaining three cases. The parameter set of residual designs of Hadamard designs is 2–(2t, t, t − 1). As we mentioned in Section 11, all 2 − (2t, t, t − 1) with t ≤ 5 have been shown to be embeddable (See [126]). Corollary 11.1 gives a solution for this parameter set with t = nk, where k ∈ {6, 7, 8, 9, 10} and n is an order of a Hadamard matrix. All These designs satisfy the three block non-embeddability condition in Theorem 9.1. Hence, 11, 13, 15, and 17 are the values of t ≤ 20 for which a non-embeddable quasi-residual 2 − (2t,t,t − 1) is not determined yet. We know that, in a 2 − (2t,t,t − 1) design, the possible size of intersection between any two blocks is at most t − 1. These results motivate us to conjecture that for t ≥ 6, if there exists a Hadamard matrix of order 4t, then there exists a non-embeddable quasi-residual design with parameters 2 − (2t,t,t − 1). Complementary designs of derived designs of a Hadamard designs are quasiresidual designs with parameters 2 − (2t − 1,t,t). According to van Lint [122], all such quasi-residual designs with t ≤ 5 are embeddable. Non-embeddable quasi-residual 2 − (2t − 1,t,t) designs are obtained for 6 ≤ t ≤ 10 (see [113], [2]). Also Corollary 11.2 demonstrates a solution of such design with t = 2nk, with k ∈ {7, 8, 9, 10} and n is an order of a Hadamard matrix. All members of this family satisfy the three block nonembeddability condition in Theorem 9.1. We suspect that it is possible to construct a non-embeddable quasi-residual 2 − (2t − 1,t,t) design for every value of t ≥ 6 for which a Hadamard matrix of order 4t exists. In Section 12, we considered quasi-residual and quasi-derived Menon designs. Corollaries 12.1 and 12.2 give non-embeddable quasi-residual and quasi-derived Menon designs of order 4n2 h2 , where n = 3, 4 and 4h2 is an order of a regular Hadamard matrix. It will be of interest to construct more non-embeddable designs of this type. One approach that might be useful is to use Paley matrices in a similar way as in Example 12.1. Ionin’s method of constructing designs using BGW and GH matrices (especially Bush type Hadamard matrices) has led to several new families of symmetric designs. We believe that his technique may also lead to the construction of non-embeddable quasiresidual designs with parameters corresponding to these families of symmetric designs. Alraqad [2, Chapter VII] included tables listing all possible parameter sets of quasiresidual designs with k ≤ v/2 and 10 ≤ r ≤ 41 and with k > v/2 and 11 ≤ r ≤ 45. See also the tables in [3, Appendix]. These tables summarize all results (known to the authors) on quasi-residual designs with these parameter sets. They indicate the number of non-embeddable designs that has been constructed for every possible parameter set

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and include the references for these constructions. In addition, they indicate the different types of non-embeddability condition which is satisfied by these designs. Acknowledgement The first author acknowledges the hospitality and support of the Institute of Mathematics and Physics (IMPA) at Aberystwyth,Wales, where the work on this manuscript was begun. He thanks Tom McDonough and Vass Mavron for their valuable input. References [1] T.A. Alraqad, New families of non-embeddable quasi-derived designs, J. Comb. Des. 16 (2007), 263275. [2] T.A. Alraqad, Construction and non-embeddability of quasi-residual designs, Ph.D. Dissertation. Central Michigan University, (2008). [3] T.A. Alraqad and M.S. Shrikhande, An overview of embedding problems of quasi-residual designs, J. Statist. Theory and Practice 3 (2009), 319-347. [4] T.A. Alraqad and M.S. Shrikhande, New families of non-embeddable quasi-residual Menon designs, J. Comb. Des. 17 (2008), 53-62. [5] E.F. Assmus and J.D. Key, Designs and their codes, Cambridge University Press, Cambridge, 1992. [6] M. Behbahani and H. Kharaghani, On a new class of productive regular Hadamard matrices, Discr. Math. 306 (2006), 3042-3050. [7] T. Beth, D. Jungnickel, and H. Lenz, Design Theory, Volumes I and II, Cambridge University Press, Cambridge, 1999. [8] A. Beutelspacher and U. Rosenbaum, Geometric authentication systems, Ratio Mathematica 1 (1990), 39-50. [9] A. Beutelspacher and U. Rosenbaum, Projective Geometry, Cambridge University Press, Cambridge, 1998. [10] V.K. Bhargava and J.M. Stein, (v, k, λ) configurations and self-dual codes, Inform. and Control 28 (1975), 352-355. [11] K.N. Bhattacharya, A new balanced incomplete block design, Sci. Culture 9 (1944), 508. [12] R.C. Bose, On the construction of balanced incomplete block designs, Ann. Eugenics 9 (1939), 353399. [13] R.C. Bose and S.S. Shrikhande, Baer subdesigns of symmetric balanced incomplete block designs, in: S. Ikeda et al., eds. Essays in Probability and Statistics, Shinko Tsusho, Tokyo (1976), 1–16. [14] R.C. Bose, S.S. Shrikhande and N.M. Singhi, Edge regular multigraphs and partial geometric designs with an application to the embedding of quasi-residual designs, Academia Nationale Dei Lincei, Atti del Convegni Lincei 17 (1976), 49-81. [15] C. Bracken, New classes of self-complementary codes and quasi-symmetric designs, Des. Codes Cryptog. 41 (2006), 319-323. [16] W.G. Bridges, A (66, 26, 10) design, J. Comb. Theory (A) 35 (1983), 360. [17] A.E. Brouwer, An infinite series of symmetric designs, Math. Centrum Amsterdam Report, ZW 136/80(1983). [18] R.H. Bruck, Difference sets in a finite group, Trans. Amer. Math. Soc. 78 (1955), 464–481. [19] R.H. Bruck and H.J. Ryser, The non-existence of certain finite projective planes, Canad. J. Math., 1 (1949), 88-93. [20] K.A. Bush, Unbalanced Hadamard matrices and finite projective planes of even order, J. Comb. Th. (A), 11 (1971), 38-44. [21] S.A. Camtepe and B. Yener. Combinatorial design of key distribution mechanisms for wireless sensor networks, In: Proceedings of 9th European Symposium On Research in Computer Security (ESORICS ’04), 2004. [22] S. Chowla and H.J. Ryser, Combinatorial problems, Canad. J. Math., 2 (1950), 93-99. [23] C.J. Colbourn and J.H. Dinitz, The CRC Handbook of Combinatorial Designs, Second Edition, Chapman & Hall/CRC Press, Boca Raton, 2007. [24] C.J. Colbourn, J.H. Dinitz, D.R. Stinson, Applications of combinatorial designs to communications, cryptography, and networking, in: J.D. Lamb, D.A. Preece (Eds.), Surveys in Combinatorics, London

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Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-404

Codes and modules associated with designs and t-uniform hypergraphs Richard M. Wilson California Institute of Technology Abstract. The Smith normal form and the invariant factors of an integer matrix are introduced. We give selected examples of how invariant factors, a chain of linear codes, and self-dual codes have appeared and been applied in the theory of combinatorial designs. In the latter part of these notes, we are concerned with diagonal forms of various incidence matrices arising from designs and uniform hypergraphs. Results on diagonal forms of such matrices can be applied to a certain zero-sum Ramsey-type problem. Keywords. incidence matrices, Smith normal form, hypergraphs, p-rank, t-uniform hypergraphs

1. Introduction These notes survey some appearances of Smith normal form (or invariant factors, or elementary divisors) and diagonal forms of integer matrices that arise in the theory of combinatorial designs. Background information on Smith form is included, occasionally for its own interest. We are concerned with p-ary codes for primes p that are generated by or arise from incidence matrices. It is of particular interest when some of these codes can be proved to be self-dual with respect to a particular inner product, as Witt’s Theorem gives a necessary condition for the existence of such codes. The simplest way to get a p-ary code from an integer matrix A is to take its row (or column) space modulo a prime p. The dimension of this code is the p-rank of A, and it is equal to the number of invariant factors of A that are not divisible by p. But other p-ary codes may be constructed when various powers of p exactly divide the invariant factors. The invariant factors, and hence the rank modulo p, of a matrix A do not change on permutation of the rows and columns (or transposition) of a matrix. Thus they do not depend on the ordering of the vertices when A is the adjacency matrix of a graph G, or on the ordering of points and blocks of some incidence structure S, etc. The invariant factors of the adjacency matrix of a graph G, or the incidence matrix of S, are therefore invariants of G or S, respectively; they are the same for two isomorphic graphs or incidence structures. So, for example, two graphs may be shown to be nonisomorphic by showing that their adjacency matrices have different invariant factors. Brouwer and Van Eijl [4] give Smith forms or p-ranks for various strongly regular graphs with the same parameters but different invariant factors. Chandler and Xiang [7] have, for example, shown that

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certain difference sets (the HKM and Lin difference sets) and their associated designs are non-isomorphic with calculations of the Smith form of their incidence matrices. We introduce Smith and diagonal forms in Section 2 and give examples in Section 3. Some properties of the invariant factors and the connection between integer solutions of systems of linear equations and diagonal forms are discussed in Sections 4 and 5. The next sections describe some classical results, starting with observations of Morris Newman [22]. Section 6 initiates our concern with a “symmetry” among the invariant factors of e.g. Hadamard matrices and the incidence matrices of symmetric designs. A chain of p-ary codes is introduced in Section 7. Section 10 presents results of Lander [19] on symmetric designs and Section 11 gives results of Blokhuis and Calderbank [2] on certain nonsymmetric designs. Our presentation of the material in these sections is more-or-less in historical order. Theorems first given in early sections may be presented in a more general form later. It is hoped that this will make the material easier to understand, though some readers may find the approach a bit repetitive. In Sections 12 and 13, we introduce inclusion matrices of t-subsets versus k-subsets, and, more generally, incidence matrices of t-subsets versus hypergraphs isomorphic to a given t-uniform hypergraph H. Diagonal forms for the inclusion matrices are described. The results are applied to the binary case of a zero-sum Ramsey-type problem introduced by Alon and Caro [1] in Section 14. In Section 15, we describe some recent joint work with Tony W. H. Wong on diagonal forms of the latter incidence matrices, in particular when t = 2 and H is a simple graph. 2. Smith and diagonal form Two integer matrices A and B of the same size are Z-equivalent when there exist unimodular matrices (square integer matrices that have integer inverses, or what is the same have determinants ±1) E and F so that EAF = B. This means that B can be obtained from A by a sequence of Z-row operations and Z-column operations (permuting rows/columns, adding an integer multiple of one row or column to another row or column, or multiplying a row or column by −1). Given A, there is a unique diagonal integer matrix D that is Z-equivalent to A such that the diagonal entries d1 , d2 , d3 , . . . are nonnegative integers and where di divides di+1 for i = 1, 2, . . . . Here ‘diagonal’ means that the (i, j)-entry of D is 0 unless i = j, but D has the same shape as A and is not necessarily square. This unique diagonal matrix is called the integer Smith normal form, or simply the Smith form, of A. (The Smith form D is unique; the unimodular matrices E and F so that EAF = D are not. See [23], Appendix C of [19], or any advanced text on modern algebra for proofs of existence and uniqueness.) The diagonal entries of the Smith form, in the order of divisibility (with all 0’s necessarily at the end) are called the invariant factors,  or theelementary  of A.  divisors 1 0 0 3 1 4 As a simple example, the Smith form of A = 4 −2 7 is D = 0 5 0 because

⎛ ⎞

0 −1 3



3 1 4 ⎝ 10 1 0 0 1 −1 −1⎠ = . 4 −2 7 21 0 5 0 0 1 −2

(1)

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The invariant factors of A are 1 and 5. We call any diagonal matrix D that is Z-equivalent to A a diagonal form for A. As we have defined them above, the number of invariant factors of a matrix A, or the number of diagonal entries of a diagonal form of A, is equal to the minimum of the number of rows and the number of columns of A. But there are numerous times when it is desirable to consider the number of invariant factors to be equal to, say, the number n of columns of a k by n matrix A even if k < n. In this case, the invariant factors si with k < i ≤ n are to be understood as 0. This agrees with the result when n − k or more rows of all zeros are appended to A (and this does not change the Z-span of the rows). To illustrate with the example in (1), ⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞ 100 0 −1 3 3 1 4 1 0 0 ⎝2 1 0⎠ ⎝4 −2 7⎠ ⎝1 −1 −1⎠ = ⎝0 5 0⎠ . 0 1 −2 0 0 0 001 0 0 0

Now the invariant factors of the middle matrix on the left are 1, 5, 0. The context should make it clear whether we are thinking of k or n as the number of invariant factors. Because (EAF )⊤ = F ⊤ A⊤ E ⊤ , it is clear that the invariant factors of A and A⊤ are the same, save possibly for trailing 0’s in case that A is not square. The Smith form of A may be obtained from any diagonal form D of A by a sequence of operations that replace two diagonal entries a and b by g = GCD(a, b) and ab/g = LCM(a, b). This can be effected by Z-row and column operations, as we indicate by the sequence of 2 by 2 matrices below. First write g = sa + tb with s and t integral.









a tb a sa+tb = g a g 0 g g 0 a0 → → → → → 0b 0 b 0 b −ab/g 0 −ab/g 0 0 ab/g We will be especially interested in the p-contributions to the invariant factors for certain primes p. These may be called the invariant p-factors. If the invariant p-factors are known for all primes p, then, of course, the invariant factors are known. The pcontributions to the diagonal entries of any diagonal form, when arranged in increasing order (though with any 0’s at the end), are the invariant p-factors. Smith form may be considered over any principal ideal domain D. When D is the p-adic integers, or the ring of rational numbers a/b where, when in lowest terms, p | b, two integer matrices will be D-equivalent if and only if they have the same invariant p-factors as defined above.

3. Some examples The invariant factors of “random” square integer matrices (including e.g. n by n matrices of 0’s and 1’s) tend to be 1’s followed by a single large factor. (The product s1 s2 · · · sn−1 is the GCD of all (n−1) by (n−1) determinants—see Section 4—and this GCD is often 1.) But matrices arising from combinatorial designs often have more interesting invariant factors. Projective planes are defined in Section 6 below. The incidence matrix N of a projective plane of order n is a v by v (0, 1)-matrix, where v = n2 +n+1. The invariant factors of some projective planes are given below; the number in parentheses is the order n

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of the plane. (Here and below in similar instances, the exponents denote the multiplicities of the integers as invariant factors. The exponents sum to the size of the matrix.) 129 , 727 , 561 1 , 29 , 49 , 826 , 721 137 , 318 , 935 , 901 141 , 310 , 939 , 901

PG2 (7) PG2 (8) PG2 (9) H(9)

28

Here PG2 (q) denotes the Desarguesian projective plane of prime power order q. We have used H(9) to denote any one of the Hall plane, the dual Hall plane, and the Hughes plane. This small table is extracted from a larger one of Moorehouse [21]. It is of interest that the invariant factors of a related matrix are somewhat easier to understand, and a symmetry is more readily visible. Let A be obtained by adjoining a row and column of all ones to the incidence matrix N , except for an entry of n + 1 where the new row and column meet; cf. (11). Then invariant factors of these matrices are given below; see the proof of Theorem 13. bordered PG2 (7) bordered PG2 (8) bordered PG2 (9) bordered H(9)

129 , 729 1 , 29 , 49 , 828 137 , 318 , 937 141 , 310 , 941 28

  Following is the 62 by 63 inclusion matrix of the 2-subsets versus the 3-subsets of a 6 6-set, called W23 in Section 12. The diagonal entries of one diagonal form are 19 , 25 , 31 . ⎛1 ⎜ 11 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎜0 ⎜0 ⎝ 0 0

1 0 0 1 1 0 0 0 0 0 0 0 0 0 0

0 1 0 1 0 1 0 0 0 0 0 0 0 0 0

0 0 1 0 1 1 0 0 0 0 0 0 0 0 0

1 0 0 0 0 0 1 1 0 0 0 0 0 0 0

0 1 0 0 0 0 1 0 1 0 0 0 0 0 0

0 0 1 0 0 0 0 1 1 0 0 0 0 0 0

0 0 0 1 0 0 1 0 0 1 0 0 0 0 0

0 0 0 0 1 0 0 1 0 1 0 0 0 0 0

0 0 0 0 0 1 0 0 1 1 0 0 0 0 0

1 0 0 0 0 0 0 0 0 0 1 1 0 0 0

0 1 0 0 0 0 0 0 0 0 1 0 1 0 0

0 0 1 0 0 0 0 0 0 0 0 1 1 0 0

0 0 0 1 0 0 0 0 0 0 1 0 0 1 0

0 0 0 0 1 0 0 0 0 0 0 1 0 1 0

0 0 0 0 0 1 0 0 0 0 0 0 1 1 0

0 0 0 0 0 0 1 0 0 0 1 0 0 0 1

0 0 0 0 0 0 0 1 0 0 0 1 0 0 1

0 0 0 0 0 0 0 0 1 0 0 0 1 0 1

0 0 ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎟ 1⎟ ⎟ 0⎟ 0⎟ ⎟ 0⎠ 1 1

⎞

The invariant factors of the adjacency matrices of the triangular graphs T (n) (the line graphs L(Kn ) of complete graphs) are as follows; see Brouwer and Van Eijl [4]. (1)n−2 , (2)(n−2)(n−3)/2 , (2n − 8)n−2 , ((n − 2)(n − 4))1

if n ≥ 4 is even,

(1)n−1 , (2)(n−1)(n−4)/2 , (2n − 8)n−2 , (2(n − 2)(n − 4))1

if n ≥ 5 is odd.

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The graph T (n) is strongly regular and determined up to isomorphism by its parameters except when n = 8, in which case there are three other SRGs (called the Chang graphs) with the same parameters. The invariant factors of each of the three Chang graphs are 18 , 212 , 87 , 241 , and these are different from those of T (8). A Hadamard matrix of order m is an m by m matrix H with entries ±1 so that HH ⊤ = mI, i.e. such that any two distinct rows are orthogonal. An important family of Hadamard matrices may be obtained recursively by defining H0 = (1) and Hn =



Hn−1 Hn−1 Hn−1 −Hn−1



for n = 1, 2, . . . . Then Hn has order 2n and it is easy to prove, by induction on n, that Hn is a Hadamard matrix. Theorem 1 The Hadamard matrix Hn as defined above has invariant factors n

n

(1)1 , (2)n , (4)( 2 ) , (8)( 3 ) , . . . , (2n−1 )n , (2n )1 . Proof. (Sketch.) Define a sequence of matrices Ei recursively by E0 = (1) and En =



En−1 En−1 En−1 O



for n = 1, 2, . . . . Then En has order 2n , 1’s on the back-diagonal, and 0’s below; hence En is unimodular. For example, we show E3 , H3 , and their product below. ⎛1 1 ⎜1 ⎜ ⎜1 ⎜1 ⎜ ⎝1 1 1

1 0 1 0 1 0 1 0

1 1 0 0 1 1 0 0

1 0 0 0 1 0 0 0

1 1 1 1 0 0 0 0

1 0 1 0 0 0 0 0

1 1 0 0 0 0 0 0

1⎞⎛1 0 1 ⎜ 0⎟ ⎟⎜1 0⎟⎜1 ⎜ 0⎟ ⎟⎜1 0⎠⎝1 0 1 0 1

1 −1 1 −1 1 −1 1 −1

1 1 −1 −1 1 1 −1 −1

1 −1 −1 1 1 −1 −1 1

1 1 1 1 −1 −1 −1 −1

1 −1 1 −1 −1 1 −1 1

1 1 −1 −1 −1 −1 1 1

1 ⎞ −1 −1 ⎟ ⎟ 1 ⎟ −1 ⎟ ⎟ 1 ⎠ 1 −1

=

⎛8 4 ⎜4 ⎜ ⎜2 ⎜4 ⎜ ⎝2 2 1

0 4 0 2 0 2 0 1

0 0 4 2 0 0 2 1

0 0 0 2 0 0 0 1

0 0 0 0 4 2 2 1

0 0 0 0 0 2 0 1

0 0 0 0 0 0 2 1

0⎞ 0 0⎟ ⎟ 0⎟ , 0⎟ ⎟ 0⎠ 0 1

In general, one can show by induction on n, using En H n =



O 2En−1 Hn−1 , En−1 Hn−1 En−1 Hn−1

that En Hn= Dn Un where Dn is a diagonal matrix with diagonal entries 2k of multiplicity nk , in some order, and Un is lower triangular with 1’s on the diagonal. For example,

E3 H 3 = D 3 U 3 =

⎛8 0 ⎜0 ⎜ ⎜0 ⎜0 ⎜ ⎝0 0 0

0 4 0 0 0 0 0 0

0 0 4 0 0 0 0 0

0 0 0 2 0 0 0 0

0 0 0 0 4 0 0 0

0 0 0 0 0 2 0 0

0 0 0 0 0 0 2 0

0⎞⎛1 1 0 ⎜ 0⎟ ⎟⎜1 0⎟⎜1 ⎜ 0⎟ ⎟⎜1 0⎠⎝1 0 1 1 1

0 1 0 1 0 1 0 1

0 0 1 1 0 0 1 1

0 0 0 1 0 0 0 1

0 0 0 0 1 1 1 1

0 0 0 0 0 1 0 1

0 0 0 0 0 0 1 1

0⎞ 0 0⎟ ⎟ 0⎟ . 0⎟ ⎟ 0⎠ 0 1

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The matrix Un is unimodular, and En Hn Un−1 = Dn . This means that Dn is a diagonal form for Hn . 

4. Basic properties of the invariant factors In these notes, module will always mean Z-module, i.e. a module over the ring Z of integers (i.e. an abelian group). Submodules of Zn may be called lattices. Given an r by m integer matrix A, we use rowZ (A) to denote the module generated by the rows of A, a submodule of Zm ; similarly, colZ (A) will denote the module generated by the columns of A, a submodule of Zr . Matrices A and A′ (of the same size) with the same row module will have the same invariant factors, because A and A′ have the same row module if and only if A and A′ are Z-row equivalent, or what is the same, A′ = EA where E is unimodular. A similar statement holds for matrices with the same column module. Given A, it is always possible to find a Z-row equivalent matrix A′ whose nonzero rows are linearly independent (cf. Hermite normal form), or a Z-column equivalent A′′ whose nonzero columns are linearly independent. Suppose D = EAF is a diagonal form for A, where E and F are unimodular. Then A has the same row module as DF −1 ; that is, a Z-spanning set for rowZ (A) consists of the vectors d1 f1 , d2 f2 , . . . , dn fn

(2)

where fi is the i-th row of F −1 . The vectors f1 , . . . , fn form a Z-basis for Zn . The fi ’s for which di = 0 form a Z-basis for the integer vectors in the row space of A. A Z-basis for rowZ (A) consists of those vectors di fi where di = 0. The map c1 f1 + · · · + cn fn → (c1 (mod d1 ), . . . , cn (mod dn )) is a homomorphism, from Zn onto the direct sum of cyclic groups Zdi , with kernel rowZ (A); thus Zn /rowZ (A) ∼ = Zd1 ⊕ Zd2 ⊕ · · · ⊕ Zdn .

(3)

Here, of course, Z0 = Z and Z1 = {0}. As an example, a Z-basis for rowZ (A), when E, A, F, D are the matrices in (1), consists of the first two rows of DF −1 , and these are (3, 1, 4) and 5(2, 0, 3). The two vectors (3, 1, 4) and (2, 0, 3) are linearly independent over every field. The latter is not in rowZ (A), though its multiple (10, 0, 15) is. We can see that the p-rank of A (the rank of A over the field Fp ) is 2 except when p = 5. Proposition 2 Let ℓ be the LCM of the nonzero diagonal entries d1 , d2 , . . . , dn of a diagonal form for A, where n is the number of columns of A. (This is equal to the last nonzero invariant factor.) (i) If v is an integer vector in rowQ (A), then ℓv ∈ rowZ (A). (ii) Conversely, if tv ∈ rowZ (A) for every integer vector v ∈ rowQ (A), then t is a multiple of ℓ.

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Proof. Let f1 , f2 , . . . , fn be a Z-basis for Zn so that the nonzero vectors among d1 f1 , d2 f2 , . . . , dn fn , say the first r, form a Z-basis for rowZ (A). Clearly ℓ times any integer linear combination of f1 , . . . , fr is an integer linear combination of d1 f1 , . . . , dn fn , which is (i). The second assertion is similarly easy.  When A is square of order n, it follows from EAF = D that the product s1 s2 · · · sn of the invariant factors is, up to sign, the determinant of A. In general, whether A is square or not, the number of nonzero invariant factors is the rank of A. It is well known that the GCD of the determinants of all k by k submatrices of A is the product s1 s2 · · · sk . (These partial products are the determinantal divisors of A. See [23].) As an illustration of determinantal divisors, note that if a square integer matrix A is nonsingular and has invariant factors s1 , s2 , . . . , sn , then sn is the least positive integer t so that tA−1 is integral. This is a consequence of the formula A−1 =

1 Aadj det(A)

where Aadj is the classical adjoint of A, with (i, j)-entry (−1)i+j det(Aji ), and where Aji is the result of deleting row j and column i from A. The GCD of the determinants det(Aji ) is s1 s2 · · · sn−1 and det(A) = s1 · · · sn . But a much easier way to see this is simply to note that if D = EAF is the Smith form of A, then F −1 A−1 E −1 = D−1 , and tA−1 will be integral if and only if tD−1 is integral. A more general statement is a quick corollary to Proposition 2 and its version for columns. Proposition 3 Suppose an n by k integer matrix A has rank n, so that A has a right inverse over the rationals Q. Then there exists a k by n integer matrix B with AB = tI if and only if t is a multiple of the largest invariant factor sn of A. Similarly, given that B is k by n of rank n, there is an integer matrix A so that AB = tI if and only if t is a multiple of the largest invariant factor s′n of B. Proof. The relation AM = I over Q means that each column of I, and hence each rational column vector of height n, is a rational linear combination of the columns of A. By Proposition 2, tv ∈ colZ (A) for every integer vector v if and only if sn divides t. The proof of the second assertion is similar.  Proposition 4 Suppose rowZ (B) ⊆ rowZ (A), or equivalently, B = M A for some integer matrix M . Let a1 , a2 , . . . , an be the invariant factors of A, where n is the number of columns of A, and b1 , b2 , . . . , bn the invariant factors of B. Then ai divides bi for i = 1, 2, . . . , n. If B has rank n, the quotient (b1 · · · bn )/(a1 · · · an ) is the index of rowZ (B) in rowZ (A). Proof. >From (3), Zn /rowZ (A) ∼ = Za1 ⊕· · ·⊕Zan and Zn /rowZ (B) ∼ = Zb1 ⊕· · ·⊕Zbn . If rowZ (B) is a subgroup of rowZ (A), then by the third isomorphism theorem, > (Zn /rowZ (B)) (rowZ (A)/rowZ (B)) ∼ = Zn /rowZ (A).

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In particular, Za1 ⊕ · · · ⊕ Zan is isomorphic to a homomorphic image of Zb1 ⊕ · · · ⊕ Zbn . It is known, when a1 |a2 | . . . |an and b1 |b2 | . . . |bn , that this is the case if and only if ai |bi for all i = 1, . . . , n. When the groups are finite, the index of rowZ (A) in Zn is a1 · · · an and the index of rowZ (B) in Zn is b1 · · · bn . Hence rowZ (A)/rowZ (B) has order (b1 · · · bn )/(a1 · · · an ).  Since the invariant factors of a matrix are the same as those of its transpose, corollaries of Proposition 4 are that the invariant factors of AM and M1 AM2 are multiples of those of A, in their correct order. The following two propositions are interesting and they allow for alternate proofs of some material. E.g. we will sketch another proof of the first part of Proposition 4 after the proof of Proposition 5. Proposition 5 The number of invariant factors of a matrix A that are not divisible by a given integer m is the minimum Q-rank of matrices B such that B ≡ A (mod m). Proof. Suppose EAF = D where D = diag(s1 , s2 , . . . ) with si |si+1 and where s1 , . . . , sr are not divisible by m but m|si for i > r. Then A ≡ E −1 D0 F −1 (mod m) where D0 = diag(s1 , . . . , sr , 0, . . . ), and −1 E D0 F −1 has rank r. Suppose A ≡ B (mod m). Then B has the same rank as a matrix of the form D + mN with N an integer matrix. Let a1 , a2 , . . . , ar be the first r rows of D + mN . If these vectors are linearly dependent, there are integer coefficients c1 , . . . , cr with GCD equal to 1 so that c1 a1 + · · · + cr ar = 0. But ai ≡ si ei (mod m) where ei is the i-th standard basis vector, so ci si ≡ 0 (mod m) and, in particular, each ci with 1 ≤ i ≤ r is divisible by m/(m, sr ), which is greater than 1. This contradiction shows that the rank of D + mN is at least r.  For another proof of the first part of Proposition 4, first note that it is sufficient to prove for each integer m that the number of bi that are not divisible by m is at least the number of ai that are not divisible by m. (One may first consider the p-contributions to the terms of each sequence.) But if A ≡ L (mod m) where rankQ (L) = r, then B = M A ≡ M L (mod m), and M L has rank at most r. So the result is a consequence of Proposition 5. The following proposition is the simplest instance of a number of relations given by R. C. Thompson in [27], though stated there only for square matrices. Proposition 6 Let A and B be r by n and n by k integer matrices with invariant factors s1 , s2 , . . . , sn and s′1 , s′2 , . . . , s′n , respectively. Let t1 , t2 , . . . be the invariant factors of AB, with the understanding that ti = 0 if i > r or i > k. Then si s′j divides ti+j−1 for 1 ≤ i, j ≤ n. Proof. Suppose that d divides si if and only if i ≥ ℓ + 1, and that d′ divides s′i if and only if i ≥ m + 1. We will show that dd′ divides ti for all i ≥ ℓ + m + 1. This will prove the proposition (details omitted). By Proposition 5,

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A = A0 + dA1

and B = B0 + d′ B1

for some integer matrices A0 , A1 , B0 , B1 where the Q-ranks of A0 and B0 are ℓ and m, respectively. Then AB ≡ A0 (B0 + d′ B1 ) + dA1 B0

(mod dd′ ).

(4)

Since rankQ (A0 ) = ℓ, it follows rankQ (A0 (B0 + d′ B1 )) ≤ ℓ, and of course rankQ (dA1 B0 ) ≤ m, so the Q-rank of the right hand side of (4) is at most ℓ + m.  Proposition 5 tells us that the number of ti ’s not divisible by dd′ is at most ℓ + m.

5. Solutions of linear equations in integers Diagonal forms are related to solutions of systems of linear equations or congruences in integers. This, in fact, was the topic of H. J. S. Smith’s original paper [26] on the subject. Let A be an r by m integer matrix. Suppose EAF = D where E and F are unimodular and D is diagonal with diagonal entries d1 , d2 , . . . . The system Ax = b is equivalent to (AF )(F −1 x) = b, and this has integer solutions x if and only if (AF )z = b has an integer solution z. This in turn will have integer solution if and only if EAF z = Eb, or Dz = Eb, has integer solutions. In other words, if we let ei denote the i-th row of E, the system Ax = b has integer solutions if and only if ei b ≡ 0 (mod di )

for i = 1, 2, . . . , r.

(5)

As a simple example,

10 21

⎞ ⎛



0 −1 3

3 1 4 ⎝ 100 ⎠ 1 −1 −1 = 4 −2 7 050 0 1 −2

and so the system of equations



3x + y + 4z = a 4x − 2y + 7z = b

has an integer solution if and only if a ≡ 0 (mod 1) (that is, a is an integer) and 2a + b ≡ 0 (mod 5). The following lemma has numerous applications. We will not need it here, but include a proof because it uses diagonal form. Lemma 7 Given a rational matrix A and a column vector b, the system Ax = b has an integer solution x if and only if for any rational row vector y, yA integral implies yb is an integer.

(6)

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Proof. The ‘only if’ direction is easy: If Ax = b with x integral and yA is integral, then yb = y(Ax) = (yA)x is an integer. If Ax = b has no solutions, then there is a row vector ei and integer di , as in (5) above, so that ei b ≡ 0 (mod di ). For simplicity, assume all di are nonzero, and let y = d1i ei . Then yb is not an integer, but yA = (0 . . . , 0,

1 1 , 0, . . . , 0)EA = (0 . . . , 0, , 0, . . . , 0)DF −1 , di di

which is the i-th row of F −1 and so is an integer vector.



Suppose EA = DU for any E with rows ei , square or not, where D is diagonal with diagonal entries di , and U is integral. Then the conditions ei b ≡ 0 (mod di ) are clearly necessary for the existence of an integer solution x of Ax = b. Lemma 8 is proved in [32]. We refer to it at one point in Section 13. Lemma 8 Let A be an r by m matrix. Suppose EA = DU where E, D, and U are integer matrices with E unimodular and D diagonal. If the conditions ei b ≡ 0 (mod di ) are sufficient for the existence of an integer solution x of Ax = b, then D, with the addition or deletion of rows or columns of all 0’s if necessary to make it r by m, is a diagonal form for A. Also, the rows of U corresponding to nonzero diagonal entries of D are linearly independent over all fields.

6. Square incidence matrices The first two theorems are from Newman [22]. Theorem 9 applies to a large class of matrices that may be found in the theory of combinatorial designs: A weighing matrix W is a square matrix with entries from {0, 1, −1} such that W W ⊤ = mI for some integer m, i.e. so that any row has m nonzero entries and any two distinct rows are orthogonal. Instances include Hadamard matrices, as defined in the previous section, and conference matrices, which are n by n matrices C with 0’s on the diagonal, and entries ±1 otherwise, so that CC ⊤ = (n − 1)I. Theorem 9 Suppose A is an n by n integer matrix such that AA⊤ = mI for some integer m. Let s1 , s2 , . . . , sn be the invariant factors of A. Then si sn+1−i = m for i = 1, 2, . . . , n. Proof. The equation AA⊤ = mI means that mA−1 = A⊤ is an integer matrix. Then the invariant factors of mA−1 are m/sn , m/sn−1 , . . . , m/s2 , m/s1 . To see this, suppose EAF = D for some unimodular matrices E and F , where D = diag(s1 , s2 , . . . , sn ) is the Smith form of A, with diagonal entries s 1 | s2 | · · · | sn .

(7)

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Then F −1 (mA−1 )E −1 = mD−1 . That is, mD−1 is a diagonal form for mA−1 . It is not necessarily the Smith form, since the diagonal element m/si+1 divides m/si and not the other way around. But the invariant factors of mA−1 in the correct order will be ?m?m m ?? m ?? ? . ... ? sn sn−1 s2 s1

(8)

But mA−1 = A⊤ , and A⊤ has the same invariant factors as A, so the factors in (7) are, by the uniqueness of the Smith form, identical to those in (8). A second proof: By Proposition 6, si sn−i+1 divides the n-th

n invariant factor of mI, which is m. The equation mn = det(mI) = det(A)2 = i=1 si sn−i+1 forces si sn−i+1 = m for each i = 1, 2, . . . , n.  As in Section 3, a Hadamard matrix of order n is an n by n matrix H, with entries +1 and −1 only, so that HH ⊤ = nI. It is known that the existence of a Hadamard matrix of order n implies n = 1, 2, or 4t for some integer t. Two Hadamard matrices of the same order may have different invariant factors. For example, the two Hadamard matrices of order 36 in Sloane’s table [25] called “had.36.pal2” and “had.36.will” have invariant factors 11 , 217 , 1817 , 361

and

11 , 215 , 64 , 1815 , 361 ,

respectively. This computation shows the matrices are not equivalent (under row and column permutations and/or multiplication of rows or columns by −1). Theorem 10 If H is a Hadamard matrix of order n = 4t with t squarefree, then the invariant factors of H are (1)1 ,

(2)2t−1 ,

(2t)2t−1 ,

(4t)1 .

Proof. By Theorem 9, the invariant factors si of H satisfy si sn+1−i = n = 4t. Since the entries of H are ±1, it is clear that s1 = 1, and since the 2-rank of H is 1, all invariant factors of H are even except for the smallest, s1 . For i ≤ n/2, si divides sn+1−i , so s2i divides 4t. Since t is squarefree, we conclude that si divides 2, and so is equal to 2 for i = 2, 3, . . . , n/2. The theorem follows.  Theorem 9 implies that a conference matrix of order n = 2t with n − 1 squarefree has invariant factors (1)t , (n − 1)t . (If the order n of a conference matrix is > 1, then it must be even.) An extension of Theorem 9 is given below. Theorem 11 Suppose A is an n by n integer matrix such that AU A⊤ = mV for some positive integer m, where U and V are square matrices of order n. Let s1 , s2 , . . . , sn be the invariant factors of A. (i) If det(U ) = det(V ) and these determinants are relatively prime to m, then si sn+1−i = m for i = 1, 2, . . . , n. (ii) If det(U ) and det(V ) are not divisible by a prime p, then the p-contribution to si sn+1−i is the same as the pcontribution to m for i = 1, 2, . . . , n.

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Proof. We have m det(V )A−1 = U A⊤ (det(V )V −1 ). (We have multiplied by det(V ) so that all matrices on the right are integral.) As in the proof of Theorem 9, the invariant factors of m det(V )A−1 are m det(V )/sn+1−i , i = 1, 2, . . . , n, in that order. But by Proposition 4, the invariant factors of U A⊤ (det(V )V −1 ) are multiples of those of A⊤ , which are s1 , . . . , sn in that order. That is, m det(V )/sn+1−i is a multiple of si , or what is the same, si sn+1−i divides m det(V ). An alternate proof of si sn+1−i | m det(V ): Proposition 4 says that sn+1−i divides the (n + 1 − i)-th invariant factor s′n+1−i of U A⊤ and Proposition 6 says that si s′n+1−i divides the n-th invariant factor of mV , which is a divisor of m det(V ). For part (i), we take determinants of both sides of AU A⊤ = mV to find det(A)2 = n m . Hence the invariant factors s1 , s2 , . . . , sn of A are prime to det(V ), and so si sn+1−i divides m for i = 1, 2, . . . , n. The equation det(A)2 = mn then forces equality for each i. For part (ii), the p contribution to si sn+1−i will divide the p-contribution to m for i = 1, 2, . . . , n. The equation det(A)2 det(U ) = mn det(V ) then forces equality for each i.  A 2-(v, k, λ)-design consists of a v-set X (of points) and a family B of k-subsets (called blocks) of X so that any two distinct points are contained in exactly λ blocks. We usually assume 2 ≤ k ≤ v − 2. For background on designs, and proofs of the observations of the next two paragraphs, see Chapter 19 of [20]. The incidence matrix N of such a design is the v by b matrix (here b = |B| = λv(v − 1)/(k(k − 1)) is the number of blocks) with rows indexed by the elements of X, columns indexed by the elements of B, and where  1 if x ∈ B, N (x, B) = 0 otherwise. It is well known that N N ⊤ = (r − λ)Iv + λJv

(9)

where r = λ(v − 1)/(k − 1) is the number of blocks that contain any given point. Also Jv N = kJvb and N Jb = rJvb . Here Jm denotes the m by m matrix of all 1’s and Jst the s by t matrix of all 1’s. For later reference we note that 

(r − λ)Iv + λJv

−1

=

1 λ (Iv − Jv ) r−λ rk

(10)

when v > k (which implies r > λ). Equation (10) can be checked by direct computation. When |X| = |B|, i.e. v = b, the design is said to be a (v, k, λ)-symmetric design. Here the incidence matrix N is square of order v, and is nonsingular since N N ⊤ is invertible. We have r = k and the relation λ(v − 1) = k(k − 1). Then N N ⊤ = nI + λJ where n = r − λ = k − λ. A projective plane of order n is a (n2 + n + 1, n + 1, 1)symmetric design. (But the order of a projective plane is not the same as the order, or size, of its incidence matrix.) Two square symmetric matrices B and C are said to be rationally congruent when there exists a nonsingular matrix A so that ABA⊤ = C. The Hasse-Minkowski The-

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orem gives necessary and sufficient conditions for two rational matrices B and C to be rationally congruent. If there exists a (v, k, λ)-symmetric design, then the equation N N ⊤ = nI +λJ means that the v by v matrices I and nI +λJ are rationally congruent. The following classic theorem may be derived from the Hasse-Minkowski Theorem (see Section 10.4 of [13]), though more elementary proofs are known. Theorem 12 (Bruck-Ryser-Chowla) If there exists a (v, k, λ)-symmetric design with v odd, then the equation x2 = ny 2 + (−1)(v−1)/2 λz 2 , where n = k − λ, has a solution in integers x, y, z, not all zero. We can say a few simple things about the invariant factors s1 , s2 , . . . , sv of the incidence matrix N of a symmetric design in general. The equation (9) implies | det(N )| = s1 s2 · · · sv = kn(v−1)/2 . We have N −1 = N ⊤ (nI + λJ)−1 =

1 ⊤ λ N − J. n nk

The smallest integer t such that tN −1 is integral  is sv = nk/(k, λ). It is easy to see that 1 0 there are 2 by 2 submatrices of N of the form 1 1 , and this implies s1 = s2 = 1.

Theorem 13 (Deretsky [8]) Let N be the incidence matrix of a (v, k, λ)-symmetric design where k and λ are relatively prime, and write n = k − λ. The invariant factors of N satisfy s1 = s2 = 1,

si sv+2−i = n

for i = 3, 4, . . . , v − 1,

and sv = nk.

Proof. Let N be the incidence matrix of a (v, k, λ)-symmetric design. When (k, λ) = 1, we have sv = nk, and so s1 s2 · · · sv−1 = n(v−3)/2 . Let t1 , t2 , . . . , tv+1 be the invariant factors of the bordered matrix

A=

N

1 .. . 1

λ ··· λ

.

(11)

k

Let D = diag(1, 1, . . . , 1, −λ), of order v + 1. It may be checked that ADA⊤ = nD. If (k, λ) = 1, then (n, λ) = 1, and by Theorem 11(i), ti tv+2−i = n for all i = 1, 2 . . . , v + 1. We now relate the invariant factors s1 , s2 , . . . , sv of N to those of A. The column module of N contains a constant column, say c11, if and only if c is a multiple of k. This is because the columns of N are linearly independent and sum to the vector of all k’s. So the column module of

R.M. Wilson / Codes and Modules Associated with Designs and t-uniform Hypergraphs

[N, 1 ] =

N

1 .. .

417

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1 contains the column module of N as an index k submodule. By Proposition 4 applied to columns, the invariant factors s′1 , . . . , s′v of [N, 1 ] are divisors of s1 , . . . , sv , respectively, while the product s′1 · · · s′v is (s1 · · · sv )/k. Since sv = kn and k is relatively prime to s1 , . . . , sv−1 , the invariant factors of [N, 1 ] must be s1 , s2 , . . . , sv−1 , n. Or, since there are v + 1 columns, we may consider the invariant factors of [N, 1 ] to be s1 , s2 , . . . , sv−1 , n, 0. Since the row module of [N, 1 ] is contained in the row module of A, t1 , t2 , . . . , tv+1 divide s1 , s2 , . . . , sv−1 , n, 0, respectively. Because ti tv+2−i = n, and clearly t1 = 1, we have tv+1 = n. Because s1 s2 · · · sv−1 n = n(v−1)/2 = t1 t2 · · · tv , we conclude that ti = si for i = 1, 2, . . . , v − 1. 

7. A chain of codes A p-ary linear code of length n is a subspace C of the vector space Fnp of ordered ntuples of elements of the field Fp of p elements. Here p is a prime, and we normally think of members of C and Fnp as row vectors. All codes in these notes will be linear codes over a prime field. Given an r by n integer matrix A, we may consider the rows as vectors in Fnp . The row space rowp (A) of A over Fp is, of course, a p-ary linear code; C ⊥ is the null space of A over Fp . Multiplying a matrix on the right or left by a unimodular matrix does not change its rank modulo p, so the dimension of C = rowp (A) is the rank modulo p of a diagonal form D of A, and this is the number of diagonal entries of D that are not divisible by p. Given A, we define, for any prime p and nonnegative integer i, Mi (A) = {x ∈ Zn : pi x ∈ rowZ (A)}. We have M0 (A) = rowZ (A) and M0 (A) ⊆ M1 (A) ⊆ M2 (A) ⊆ . . . . Let Ci (A) = πp (Mi (A)) where πp is the homomorphism (projection) from Zn onto Fnp given by reading all coordinates modulo p. Then each Ci (A) is a p-ary linear code of length n; and C0 (A) = rowp (A). Clearly, C0 (A) ⊆ C1 (A) ⊆ C2 (A) ⊆ . . . .

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Theorem 14 Let D be a diagonal form for A, with diagonal entries d1 , d2 , . . . , dn where n is the number of columns of A. The dimension of the p-ary code Cj (A) is the number of diagonal entries di that are not divisible by pj+1 . Proof. Let f1 , f2 , . . . , fn be a Z-basis for Zn so that rowZ (A) is the Z-span of d1 f1 , d2 f2 , . . . , dn fn . An integer vector a1 f1 + · · · + an fn is in rowZ (A) if and only if ai ≡ 0 (mod di ) for every i, so pj (c1 f1 + · · · + cm fm ) ∈ rowZ (A) if and only if pj ci ≡ 0 (mod di ). If pj+1 divides di , then this congruence implies ci ≡ 0 (mod p); but if the p-contribution to di is at most pj , then there exist values of ci ≡ 0 (mod p) for which pj ci ≡ 0 (mod di ), so fi , when read modulo p, is in Cj (A). It is now clear that the set of fi so that pj+1 does not divide di is a basis for Cj (A).  The following proposition is included as an application of the codes Ci discussed in this section, though various other proofs are possible (e.g. a very quick proof can be obtained by consideration of Smith form over the domains D mentioned at the end of Section 2). Proposition 15 Let L and M be integer matrices with L square so that LM is defined. Suppose det(L) is relatively prime to p. Then the invariant p-factors of LM are the same as those of M . Proof. We will show Ci (LM ) = Ci (M ) for all i. Let d = det(L) and let d′ be a multiple of d so that d′ ≡ 1 (mod p). First, since the rows of LM are integer linear combinations of the rows of M , it is clear that a ∈ Mi (LM ) implies a ∈ Mi (M ) and so Ci (LM ) ⊆ Ci (M ). Suppose a ∈ Mi (M ); say pi a = cM where c is an integer vector. Then pi d′ a = c(d′ L−1 )(LM ), and c(d′ L−1 ) is an integer vector, so pi d′ a ∈ Mi (LM ). But d′ a ≡ a (mod p). 

8. An example of a chain of codes: the Reed-Muller codes The Hadamard matrix Hn introduced in Section 3 can be described as follows. Let the vectors in {0, 1}n be v1 , v2 , . . . , v2n in lexicographical order. For example, v1 = (0, 0), v2 = (0, 1), v3 = (1, 0), v4 = (1, 1) when n = 2. Here we wish to consider the elements of {0, 1} as integers. Then one can prove by induction on n that the entry in row i and column j of Hn is Hn (i, j) = (−1)vi ,vj  . If vi = (a1 , a2 , . . . , an ) and we introduce the linear functional ℓi (x1 , . . . , xn ) = a1 x1 + · · · + an xn , then vi , vj = ℓi (vj ) and the i-th row of Hn can be written as 

 (−1)i (v1 ) , (−1)i (v2 ) , . . . , (−1)i (v2n ) .

(13)

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The r-th order Reed-Muller code of length 2n , 0 ≤ r ≤ n, is the binary linear code whose elements (codewords) are   π2 f (v1 ), f (v2 ), . . . , f (v2n )

as f ranges over all multilinear polynomials f (x1 , x2 , . . . , xn ) of degree ≤ r with coefficients in {0, 1}. Here π2 is, as in the previous section, the projection into Fn2 that reads all the coordinates modulo 2. For example, since there are only two polynomials f of degree ≤ 0, namely 0 and 1, RM(0, n) contains two codewords—the vector of all 1’s and the zero vector. There are



n n n + ··· + + d(r, n) = 1 + r 2 1 multilinear monomials g(x1 , . . . , xn ) (i.e. in which each variable appears with exponent   0 or 1) of degree ≤ r, and the corresponding vectors π2 g(v1 ), . . . , g(v2n ) form a basis for RM(r, n) over F2 , which means that d(r, n) is the dimension of RM(r, n). See Section 5.2 of [3]. Theorem 16 The code Cr (Hn ) is equal to RM(r, n). Proof. We will show  that if g(x) =xi1 xi2 · · · xik is a monomial of degree k, then the integer vector 2k g(v1 ), . . . , g(v2n ) is a signed sum of 2k rows of Hn . In fact, this is easy. For notational convenience, assume g(x) = x1 x2 · · · xk . For x1 , . . . , xk ∈ {0, 1},      2k x1 x2 · · · xk = 1 − (−1)x1 1 − (−1)x2 · · · 1 − (−1)xk

= 1 − (−1)x1 − · · · + (−1)x1 +x2 + · · · + (−1)n (−1)x1 +x2 +···+xk .

Just note that if any xi is 0, then the expressions on  either side of the  first equal sign evaluate to 0; otherwise both sides are 2k . Thus 2k g(v1 ), . . . , g(v2n ) is a signed sum of vectors of the form (13) for ℓ = 0, ℓ = x1 , . . . , ℓ = x1 + x2 , . . . , ℓ = x1 + · · · + xk . In summary, if f (x) is a multilinear polynomial with integer coefficients and  de  gree ≤ r, then 2r f (v1 ), . . . , f (v2n ) ∈ rowZ (Hn ) and so π2 f (v1 ), . . . , f (v2n ) is in Cr (Hn ). Then RM(r, n) ⊆ Cr (Hn ). Theorems 1 and 14 show that the dimension of Cr (Hn ) is d(r, n), so the codes are equal. 

9. Self-dual codes; Witt’s theorem A p-ary linear code C is self-orthogonal when C ⊆ C ⊥ , and self-dual when C = C ⊥ . A self-dual code of length n has dimension n/2. Theorem 17 If there exists a self-dual p-ary code of length n, where p is an odd prime, then (−1)n/2 is a square in Fp .

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Proof. Let C be a self-dual p-ary code of length n (and dimension n/2). Then C = rowp (G) for some n/2 by n matrix G over Fp that satisfies GG⊤ = O. By row operations and permutation of columns if necessary, we may assume

G=

I

A

where both I and A are square. The equation GG⊤ = O means that AA⊤ = −I; hence det(A)2 = (−1)n/2 .  This says nothing if n ≡ 0 (mod 4) or if p ≡ 1 (mod 4), because this condition is always true. But when n ≡ 2 (mod 4) and p ≡ 3 (mod 4), there are no self-dual p-ary codes of length n. Theorem 18 If there exists a conference matrix of order n ≡ 2 (mod 4), then n − 1 is the sum of two squares. More generally, if there is a square integer matrix A of order n ≡ 2 (mod 4) so that AA⊤ = mI, then m is the sum of two squares. Proof. An integer m is the sum of two squares if and only if no prime p ≡ 3 (mod 4) divides the square-free part of m. If an odd prime p divides the squarefree part of m, Theorem 9 gives us a self-dual p-ary code of length n ≡ 2 (mod 4) and Theorem 17 implies that −1 is a square in Fp , which implies p ≡ 1 (mod 4).  We may use a symmetric nonsingular matrix U over a field Fp with p odd to introduce a new inner product ·, · U for row vectors in Fp n , namely a, c U = aU c⊤ . For a linear p-ary code C ⊂ Fnp , the U -dual code of C is C U = {a : a, c U = 0 for all c ∈ C}. In the theory of vector spaces equipped with quadratic forms, a p-ary code is said to be totally isotropic with respect to U when C ⊆ C U . When U = I, totally isotropic is the same as self-orthogonal. We may call C self-U -dual, or say that C is self-dual with respect to U , when C = C U . There are a number of theorems concerning quadratic forms to which Witt’s name is attached. See Appendix B of Lander [19]. The result that we need is Theorem B.8 of [19] and Theorem 29.2 in [20]. Theorem 19 (Witt) Given a symmetric nonsingular matrix U of order n over Fp , p odd, there exists a p-ary code of length n that is self-dual with respect to U if and only if (−1)n/2 det(U ) is a square in Fp . Proof. (Of the ‘only if’ part.) Let G be a generating matrix for such a code. So G is an n/2 by n matrix of p-rank n/2 such that GU G⊤ = O over Fp Over a field of characteristic = 2, every symmetric matrix is congruent to a diagonal matrix, i.e. there is

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a nonsingular matrix V of order n so that V U V ⊤ = D where D is diagonal. Then (GV −1 )D(GV −1 )⊤ = O. Because GV −1 has rank n/2, some set of n/2 columns is linearly independent over Fp and, for notational convenience, we may assume that the first n/2 columns are independent. That is, GV −1 =

A1

A2

where A1 is nonsingular. And then

O

=

A1

D1

O

A⊤ 1

O

D2

A⊤ 2

A2

where D has been written as the direct sum of diagonal matrices D1 and D2 of orders ⊤ n/2 n/2. This means A1 D1 A⊤ det(U ) is a square 1 = −A2 D2 A2 . The result that (−1) in Fp follows when we take determinants of both sides of this last equation, and use the fact that det(U ) differs by a square factor from det(D) = det(D1 ) det(D2 ).  (The theorem and the proof remain valid when Fp is replaced by any field F of odd characteristic and we define codes over F.) Theorem 20 Suppose A is an n by n integer matrix such that AU A⊤ = pe V for some integer m, where U and V are square matrices with determinants relatively prime to p. Then Ce (A) = Fnp and Cj (A)U = Ce−j−1 (A)

for j = 0, 1, . . . , e − 1.

In particular, if e = 2f + 1, then Cf (A) is a self-U -dual p-ary code of length n. Proof. Let x and y be integer vectors such that πp (x) ∈ Cj (A) and πp (y) ∈ Ce−j−1 (A). This means pj (x + pa1 ) = z1 A and pe−j−1 (y + pa2 ) = z2 A for some integer vectors z1 , z2 , a1 , and a2 . Then pe−1 x, y = pe−1 xU y⊤ ≡ z1 AU A⊤ z⊤ 2 ≡0

(mod pe ).

Thus x, y = 0 in Fp and we see Ce−j−1 (A) ⊆ Cj (A)U . Let s1 , s2 , . . . , sn be the invariant factors of A. By Theorem 11(ii), the pcontribution to si sn+1−i is pe . The dimension of Cj (A) is, by Theorem 14, the number ℓ of values of i such that pj+1 does not divide si . In particular, the dimension of Ce (A) is n, so Ce (A) = Fnp . In general, there are then n − ℓ values of i such that pj+1 divides si , and hence n − ℓ values of i for which pe−j does not divide si . So the dimensions of Ce−j−1 (A) and Cj (A)U are both equal to n − ℓ, and the codes are equal. 

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Theorem 21 If there exists a (v, k, λ)-symmetric design with (k, λ) = 1, then for every odd prime divisor p of the squarefree part of n, (−1)(v−1)/2 λ is a square modulo p. Proof. Let A be the v + 1 by v + 1 matrix in (11). If p2f +1 exactly divides n, then Cf (A) is a self-D-dual code of length v + 1, where D = diag(1, 1, . . . , 1, −λ). The result now follows from Theorem 19. 

10. Symmetric designs Theorem 22 (Lander [19]) Suppose there exists a symmetric (v, k, λ)-design where n = k − λ is exactly divisible by an odd power of a prime p. Write n = pf n0 (f odd) and λ = pb λ0 with (n0 , p) = (λ0 , p) = 1. Then there exists a self-dual p-ary code of length v + 1 with respect to the scalar product corresponding to  diag(1, 1, . . . , 1, −λ0 ) if b is even, U= diag(1, 1, . . . , 1, n0 λ0 ) if b is odd. Hence from Witt’s Theorem,  −(−1)(v+1)/2 λ0 is a square (mod p) (−1)(v+1)/2 n0 λ0 is a square (mod p)

if b is even, if b is odd.

Proof. Let N be the incidence matrix of a symmetric (v, k, λ)-design and let p be a prime. Assume λ = p2a λ0 where (λ0 , p) = 1 and a ≥ 0; we will explain later what to do when λ is exactly divisible by an odd power of p. Let ⎛

⎞ pa ⎜ .. ⎟ ⎜ N .⎟ A := ⎜ ⎟, ⎝ pa ⎠ pa λ 0 · · · p a λ 0 k

⎛ ⎞ 1 0 ⎜ .. ⎟ ⎜ ⎟ . U := ⎜ ⎟. ⎝ ⎠ 1 0 −λ0

The reader may verify, using the properties of N in Section 6, and the relation λ(v −1) = r(k − 1) = k(k − 1), that AU A⊤ = nU . In case λ is exactly divisible by an even power of p, we apply Theorems 19 and 20 with the matrices A and U as above. If λ is exactly divisible by an odd power of p, we apply the above case to the complement of the given symmetric design, which is a symmetric (v, v − k, λ′ )-design where λ′ = v − 2k + λ. Say λ′ = pc λ′0 where (λ′0 , p) = 1. >From λλ′ = n(n − 1), it follows that c is even and that λ0 λ′0 = n0 (n − 1) ≡ −n0

(mod p).

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There is a self-dual code with respect to diag(1, . . . , 1, −λ′0 ) if and only if there is a self-dual code with respect to diag(1, . . . , 1, n0 λ0 ), since the last coordinates differ by a nonzero square factor modulo p by (14). 

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In [19], Lander points out that the latter part of Theorem 22, the consequences of Witt’s Theorem, actually follow from the Bruck-Ryser-Chowla theorem, Theorem 12. But the existence of the self-dual codes provides a combinatorial explanation or interpretation of part of the Bruck-Ryser-Chowla Theorem, and may be of independent interest. It was the study of a hypothetical self-dual binary code of length 112 arising from a putative projective plane of order 10 that led to the proof of the nonexistence of such a plane in 1989; see Lam [17]. Lander provides a discussion of the relation between Theorem 22 and Theorem 12 in Chapter 2 of [19]. 11. Non-square incidence matrices In this section we describe some of the results of Blokhuis and Calderbank [2]. Their concern was with non-symmetric (v, k, λ)-designs with the property that the cardinality |S ∩ T | of the intersection of any two blocks is congruent to k modulo a prime p, or a power pe of p. Their work was motivated by the observation that tables of parameters for potential quasi-symmetric designs (with exactly two block intersection cardinalities other than k) included instances where the intersection cardinalities were, in fact, ≡ k (mod pe ) where pe ||(r − λ). When e is odd, they were able to prove that the existence of a design implies the existence of a self-U -dual p-ary code for some U . And sometimes this lead to the conclusion that a potential design did not exist. We describe only a small portion of their work. The reader is refered to the original paper for a complete description. The two lemmas below are from [2]. Lemma 23 Let L and M be r by k and k by r matrices, respectively, where r ≥ k. If M L = dI for some integer d, then every nonzero invariant factor of LM divides d. Proof. Let A = LM . Then A2 = L(M L)M = dLM = dA. Suppose EAF = D where E and F are unimodular and D is diagonal with diagonal entries d1 , d2 , . . . , dr . Then DF −1 E −1 D = dD. The i-th diagonal entry on the left hand  side is divisible by d2i and the i-th diagonal entry on the right hand side is ddi . Lemma 24 Let L and M be r by k and k by r matrices, respectively, where r ≥ k. Let s1 , s2 , . . . , sk and s′1 , s′2 , . . . , s′k be the invariant factors of L and M , respectively, and let t1 , t2 , . . . , tr be the invariant factors of LM (the last r − k of which must be zero, since LM has rank at most k). Then (s1 s2 · · · sk )(s′1 s′2 · · · s′k ) = t1 t2 · · · tk . Proof. As mentioned in Section 4, there exists a matrix whose nonzero rows are linearly independent that is Z-row equivalent to any given matrix; thus there exists a unimodular matrix E so that

E

L

=

L0 O

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where L0 is square of order k and has the same invariant factors as L. Similarly, there exists a unimodular matrix F so that

M

=

F

M0

O

where M0 is square of order k and has the same invariant factors as M . Then LM has the same invariant factors as

ELM F =

L 0 M0 O O

.

O

Thus the product t1 · · · tk of the invariant factors of LM is equal to det(L0 ) det(M0 ).  Here is another proof of Lemma 23. With the notation introduced above, LM and L0 M0 have the same nonzero invariant factors. If M L = dI, then

dI

=

ML =

M0

O

F −1 E −1

L0

=

M0

U

L0

O where U is the upper-left r by r principal submatrix of E −1 F −1 . That is, M0 U L0 = dI, and since a matrix commutes with its inverse, U L0 M0 = dI. This implies that the invariant factors of L0 M0 divide d. Theorem 25 ([2]) Let B be a 2-(v, k, λ) design and p an odd prime that exactly divides r − λ. Assume that |S ∩ T | ≡ k (mod p) for any two blocks S and T of the design. Finally, suppose that v is odd. If k ≡ 0 (mod p), then (−1)(v−1)/2 k is a square modulo p. If k ≡ 0 (mod p), then (−1)(v−1)/2 v is a (nonzero) square modulo p. Proof. Let N be the v by b incidence matrix of the design and let 0

1 M=

N⊤

.. .

and

M′ =

N⊤

.. . .

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0

1 1 ··· 1

1

We will show that if k ≡ 0 (mod p), then C = rowp (M ) is a self-dual p-ary code of length v + 1 with respect to U = diag(1, . . . , 1, −k), and if k ≡ 0 (mod p),

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then C ′ = rowp (M ′ ) is a self-dual p-ary code of length v + 1 with respect to U ′ = diag(1, . . . , 1, −v). Then Theorem 19 will complete the proof. It is straightforward to check that C is self-U -orthogonal or that C ′ is self-U ′ orthogonal, according to whether k ≡ 0 (mod p) or k ≡ 0 (mod p). It remains to show that the p-rank of M or M ′ is (v + 1)/2, or at least (v + 1)/2 (it cannot be greater). Let

A=

N

1 .. .

and

N⊤

B=

.

1 −λ · · · −λ Clearly 1 ≥ rankp (A) − rankp (B) ≥ 0.

(16)

Let s1 , s2 , . . . , sv and s′1 , s′2 , . . . , s′v be the invariant factors of A and B, respectively. We have AB = N N ⊤ − λJ = (r − λ)I. By Lemma 3, the invariant factors si and s′i divide r − λ, so each is divisible by at most the first power of p. Let t1 , t2 , . . . , tb+1 be the invariant factors of k

BA =

N ⊤N

.. .

.

k −λk · · · − λk −λv Here N ⊤ N is the matrix whose entry in row S and column T is the cardinality |S ∩ T | for blocks S and T , so our hypothesis means N ⊤ N ≡ kJ (mod p). The Q-rank of BA cannot exceed the number v of rows of A, but BA contains N ⊤ N , which has the same Q-rank as N N ⊤ , namely v. In summary, BA has rank v and so t1 , . . . , tv are the nonzero invariant factors. By Lemma 23, each ti divides r − λ and so cannot be divisible by more than the first power of p. By Lemma 24, (s1 s2 · · · sv )(s′1 s′2 · · · s′v ) = t1 t2 · · · tv .

(17)

Suppose k ≡ 0 (mod p). The congruence λk ≡ λv (mod p) follows from λ(v −1) = r(k − 1), and r ≡ λ (mod p), so in this case all rows of BA are constant vectors modulo p and BA has p-rank 1, i.e. v − 1 of the ti ’s are divisible by p. The p-contribution to t1 t2 . . . tv is then pv−1 , and from (16) and (17), it follows that the largest (v −1)/2 of the si ’s and the s′i ’s are divisible by p and the others are not. That is, both A and B have p-

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rank equal to (v +1)/2. If λ ≡ 0 (mod p), then rankp (N ) = rankp (B). If λ ≡ 0 (mod p), then 1 ∈ colp (N ) because the sum of all columns of N is r11 and r ≡ λ (mod p), so rankp (N ) = rankp (A). In either case, rankp (M ) ≥ rankp (N ) = (v + 1)/2. Suppose k ≡ 0 (mod p). Then BA ≡ O (mod p), so t1 , . . . , tv are divisible by p; that is, the p-contribution to t1 t2 . . . tv is pv . From (16) and (17) it follows that A has p-rank (v + 1)/2 and B has p-rank (v − 1)/2. Then rankp (M ′ ) ≥ (v + 1)/2. Note that v ≡ 0 (mod p) because otherwise colp (A) would be a self-orthogonal p-ary code of length v and dimension > v/2.  An example of feasible design parameters for which Theorem 25 proves there does not exist a corresponding design are v = 1443, k = 624, λ = 2136 and where pairs of distinct blocks meet in 246 or 273 points. The theorem is applied with p = 3. Blokhuis and Calderbank [2] also construct self-dual codes (and so obtain nonexistence results) from 2-(v, k, λ) designs in the case that for some odd power pe of a prime p, pe ||r − λ and |S ∩ T | ≡ k (mod pe ) for all pairs S, T of blocks. These codes are of the form C(e−1)/2 (M ) for various matrices M similar to those in (15).

12. The matrices of t-subsets versus k-subsets Incidence or inclusion matrices of s-subsets versus blocks arise in the theory of t-designs and in extremal set theory. By a t-vector based on X, or just a t-vector if the set X is understood, we mean a (row or column) vector whose coordinates are indexed by the t-subsets of a set X. We often use functional notation: if f is a t-vector and T a t-subset of X, then f(T ) will denote the entry of f in coordinate position T .     v For integers t, k, v with 0 ≤ t ≤ k ≤ v, let Wtk or Wtk denote the vt by kv matrix whose rows are indexed by the t-subsets of a v-set X, whose columns are indexed by the k-subsets of X, and where the entry in row T and column K is Wtk (T, K) :=



1 if T ⊆ K, 0 otherwise.

A fundamental relation holds for all 0 ≤ s ≤ t ≤ k ≤ v:

k−s Wsk . Wst Wtk = t−s

(18)

This is because (Wst Wtk )(S, K) =

 T

Wst (S, T )Wtk (T, K) =



T :S⊆T ⊆K

1=



k−s t−s

0



if S ⊆ K, otherwise.

A relatively simple induction proof of the following theorem is given in [32]. Theorem 26 For 0 ≤ t ≤ k ≤ v, there exists a Z-basis for colZ (Wtk ) consisting of columns of Wtk .

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Explicit constructions of the types of these bases have been described by e.g. Khosrovshahi and Ajoodani [14] (“non-starting blocks”, also see [16]) for all t and k. When t = 1 and k = 2, it is possible to characterize all such Z-bases. v corresponding to a set B of 2Proposition 27 For v ≥ 3, the set of columns of W12 v v subsets of X will be a Z-basis for colZ (W12 ) if and only if the graph G whose vertices are X v and edges are B is connected, has v edges (one more than a tree would), but is such that the unique cycle C in G has odd length.

Proof. (Partial proof.) Let he denote the 1-vector corresponding to an edge (2-subset) e. So if e = {x, y}, then he (x) = he (y) = 1 while he (z) = 0 otherwise. We will show that if G is as described above, then he , e ∈ B, span the column module. The remainder of the proof will be omitted. For any edge e = {x, y}, there is a walk w = (x = z0 , z1 , z2 , . . . , z = y), with edge terms e1 = {z0 , z1 }, e2 = {z1 , z2 }, . . . , e = {z−1 , z } where ℓ is odd. This is because we can always find walks w1 from x to a vertex c1 of the the cycle C, and a walk w2 from a vertex c2 of C to y, and then we can choose a walk w3 in the odd cycle C from c1 to c2 whose length has the right parity so that the concatenation w1 w3 w2 has odd length ℓ. Then he1 − he2 + he3 − · · · + he = he . This is particularly easy to understand when w is a simple path, but holds even when edge or vertex terms are repeated.  For every nonnegative integer v select and fix a v-set X v and for every choice of t with 1 ≤ t ≤ v, select and fix a family Btv of t-subsets of X v so that the corresponding v v v columns of Wt−1,t form a Z-basis for colZ (Wt−1,t ). Let the matrix Ut,k be obtained by    v  v v v deleting the rows corresponding to t-subsets in Bt from Wtk . So Ut,k is a vt − t−1     v v be the same as W0k , i.e. a 1 by kv matrix of all by kv matrix of 0’s and 1’s. Let U0k 1’s. We will drop the superscript v when there is no danger of confusion. We will use the term row-unimodular for a matrix M whose rows are linearly independent over every field. This is the same as saying the Smith form of M is [I, O] (with the zero matrix O degenerate if M is square). A row-unimodular matrix can be completed, by adjoining rows, to a unimodular matrix; a proof of this is easy since [I, O] can be completed to an identity matrix. The following lemma is from [32] but see the proof of Lemma 6 of [31] for an additional detail. A form of this lemma is present in Frankl [10]. Lemma 28 For 0 ≤ s ≤ t ≤ v − s, the matrix

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428

U0t

1 row

U1t

v − 1 rows v 

U2t Est =

s @

2

Ujt =

.. .

j=0

v 

Ust

with

v  s

− v rows

s

−



v s−1



rows

rows is row-unimodular. In particular, if 2t ≤ v, then Ett is unimodular.

>From (18), it follows that Ust Wtk = Ett

Wtk

k−s t−s

=

where Dtk is a square diagonal matrix of order

Usk . Hence, we have

Dtk v  t

,

Etk

(19)

with diagonal entries

v 1

v −v

v−1 (vt)−(t−1 ) k k − 2 (2) k−1 , . . . , (1) , , . t t−2 t−1

(20)

Theorem 29 ([30]) Let t, k, v with 0 ≤ t ≤ k ≤ v − t be given. The matrix Wtk has a diagonal form with diagonal entries as given in (20). Proof. We modify Equation (19) slightly. Let

′ = Dtk

Dtk O

and

′ Etk =

Etk V

′ ′ where V is chosen so that Etk is unimodular (e.g. we may take Etk = Ekk if 2k ≤ v). Then

Ett

Wtk

′ −1 = (Etk )

′ and now, by definition, Dtk is a diagonal form for Wtk .

′ Dtk



The question of whether there exist integer solutions x of Wtk x = λ11 is related to the existence problem for t-designs. A simple t-(v, k, λ) design consists of a set X and

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a set or multiset A of k-subsets of X so that every t-subset of X is contained in exactly λ members of A. Let u be the characteristic k-vector of a set or multiset A of k-subsets of X. This means that when A is a set, u(A) = 1 if A ∈ A and u(A) = 0 otherwise; in general, u(A) is the multiplicity of A in A. Then for a t-subset T of X, (Wtk u)(T ) =



u(A)Wtk (A) =

A∈A



1

A∈A, T ⊆A

is the number of members of A that contain T ; so (X, A) is a t-design if and only if Wtk u = λ11 where 1 is the t-vector of all 1’s. The first part of the following theorem is due to Graver and Jurkat [12] and the author [28]. See [16] for a third proof. Theorem 30 Let t + k ≤ v. (i) Necessary and sufficient conditions for the existence of  v x = λ11 are an integer k-vector x of height kv so that Wtk λ



v−i t−i



≡0

(mod



k−i ) t−i

for i = 0, 1, . . . , t.

(21)

(ii)  Necessary vand sufficient conditions for the existence of an integer k-vector x of height v 1 (mod m) are k so that Wtk x ≡ λ1

v−i λ t−i



≡0

 k − i  ) (mod GCD m, t−i

for i = 0, 1, . . . , t.

(22)

Proof. Part (i) is a consequence of part (ii) when m = 0. The necessity of the conditions follows from (18). v Note that (cf. Section 5) Wtk x ≡ λ11 (mod m) will have an integer solution x if Dtk z ≡ λEtt 1 (mod m) has an integer solution z, where  is as in (19). If e is a row of  D Ett , it is a row of Uit for some i, and then λe11 = λ v−i t−i . The corresponding entry in k−i D is t−i . The congruence Dtk z ≡ λEtt 1 (mod m) has an integer solution if and only k−i   if each congruence λ v−i t−i z ≡ t−i (mod m) has an integer solution z. The necessary and sufficient conditions for these individual congruences to have integer solutions are in (22). 

13. The matrices of t-subsets versus t-uniform hypergraphs Systems of Diophantine linear equations have come up repeatedly in work on the asymptotic existence of decompositions of complete graphs. For example, the following theorem 31 is from [29]. Theorem 31 Let G be a simple graph on k vertices and assume v ≥ k + 2. Let G be the set of all subgraphs of the complete graph Kv that are isomorphic to G. There exists a family {xH : H ∈ G} of integers xH so that for every edge e of Kv ,

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R.M. Wilson / Codes and Modules Associated with Designs and t-uniform Hypergraphs



xH = 1

H:e∈E(H)

  if and only if v2 is divisible by the number of edges of G, and v − 1 is divisible by the greatest common divisor of the degrees of the vertices of G.   The conditions that v2 is divisible by the number of edges of G, and v−1 is divisible by the greatest common divisor of the degrees of the vertices of G are necessary for the existence of a decomposition (a partition of the edges) of Kv into subgraphs isomorphic to G. Theorem 31 played an essential role in the proof given in [29] that, given G, such decompositions exist for all sufficiently large integers v satisfying these conditions. (Such decompositions may also be called G-designs.) Similar theorems, but concerning more complicated systems of equations, were needed for work on decompositions of “edge-colored complete graphs” may be found in [18] and [9]. Though it is immaterial for the original application, the hypothesis v ≥ k + 2 may be dropped as long as G is not edgeless, complete, complete bipartite, or the union of two disjoint complete graphs; see Theorem 39 below. A common generalization and extension of Theorems 30 and 31 is Theorem 32 below. Given a t-uniform hypergraph, we consider the matrix Nt or Ntv (H) whose columns are the characteristic t-vectors   of all distinct images of H under the symmetric group Sv on a v-set X. So Nt has vt rows and at most n! columns. (For most purposes, it will not matter if Nt has repeated columns.) When H is the complete t-uniform hypergraph (t) Kk , we have Nt = Wtk . Let gi denote the GCD of all entries of Wit Nt . Then, of course, Uit Nt ≡ O (mod gi ). Let Dt be the diagonal matrix whose diagonal entries are v

v

v

(g0 )1 , (g1 )v−1 , (g2 )(2)−v . . . , (gt )( t )−(t−1) .

(23)

=

(24)

Then Ett

Nt

Dt

Ft

,

where Ft , or Ft (H) is an integer matrix of the same dimensions as Nt . It turns out that Ft is row-unimodular, but this is not proved directly in [32], but rather is derived as a consequence of Theorem 32 below and Lemma 8. Theorem 32 ([32]) Let H be a t-uniform hypergraph with k vertices. If v ≥ k + t, then necessary and sufficient conditions for the existence of an integer solution x to Nt x = b for a t-vector b are Wit b ≡ 0

(mod gi )

for i = 0, 1, . . . , t.

If t = 2 and H is a graph, then N2 is the matrix of the system of equations in Theorem 31. In this case, g0 is the number of edges of H and g1 is the GCD of the degrees of H. When t = 2 and b is the vector of all 1’s, W02 b = v2 and W12 b is a vector of v − 1’s. Thus Theorem 32 implies Theorem 31.

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Theorem 33 Let H be a t-uniform hypergraph on k vertices. If v ≥ k + t, the matrix Nt has a diagonal form with diagonal entries as given in (23). Proof. This follows from (24) in a manner similar to the proof of Theorem 29.



14. A zero-sum Ramsey-type problem   Given t and k with 0 ≤ t ≤ k and a prime p so that kt ≡ 0 (mod p), what is the least integer n ≥ k so that if the t-subsets of any n-set X are colored with the elements of Fp ,   there is always some k-subset A of X such that the sum of the colors of all kt of the t-subsets of A is 0 in Fp ? The least such integer n is denoted by R(t, k; p). Such integers n exist because of the classical Ramsey’s Theorem. If n is sufficiently large, there will exist a k-subset A so that all t-subsets of A have the same color a ∈ Fp .   Our hypothesis kt ≡ 0 (mod p) means that the sum of the colors of the t-subsets in this k-subset A is 0 in Fp . The values of R(t, k; p), however, tend to be much smaller than the numbers that  arise in the classical Ramsey’s Theorem. The numbers R(t, k; p) are not defined if kt ≡ 0 (mod p), because one can color all t-subsets of an n-set with 1 and then no k-subset is such that the sum of the colors of its t-subsets is 0 in Fp .   In [6], Y. Caro proved that when kt is even, R(t, k; 2) ≤ k + t. We give the exact value of R(t, k; 2) in Theorem 34 below. In particular, Caro’s result implies R(2, k; 2) ≤ k + 2. In fact, R(2, k; 2) = k + 2, because one can color the edges of Kk+1 with 0 and 1 such that every Kk subgraph of Kk+1 contains an odd number of edges colored 1, as follows. When k is odd, color the edges of a k-cycle in Kk+1 with 1, and others with 0; any Kk subgraph contains k or k − 2 of these edges colored 1. When k is even, color the edges of a (k + 1)-cycle in Kk+1 with 1, and others with 0. n as follows. We can state the zero-sum Ramsey problem in terms of the matrices Wtk Given a coloring with Fp of the t-subsets of an n-set, let x be the t-vector such that n , with computations done in Fp , is in x(T ) is the color of the t-subset T . Then xWtk n n ). Coordinate A of xWtk is the sum of the colors of the t-subsets contained rowp (Wtk in the k-subset A. So R(t, k; p) is the least integer n ≥ k so that no vector in the p-ary n code generated n by the rows of Wtk is all-nonzero, i.e. such that there are no codewords of weight k . For p = 2, a vector has all coordinates nonzero if and only if it is equal to 1 = (1, 1, . . . , 1). So R(t, k; 2) is the least integer n ≥ k so that (1, 1, . . . , 1) is not in the n . binary code generated by the rows of Wtk The following theorem is from [33]. We repeat the proof here.   Theorem 34 When kt is even, R(t, k; 2) is equal to k + 2e where 2e is the least power of 2 that appears in the base 2 representation of t but not in the base 2 representation of k.   That kt is even implies that there are such powers of 2; see below. In particular, we have R(t, k; 2) = k + t when t is a power of 2, and R(t, k; 2) < k + t otherwise.

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n Proof. Theorem 30(ii) gives conditions under which 1 is in colm (Wab ) when a ≤ b ≤ n n − a. But here we are interested in rowm (Wtk ) when k + t ≥ n. By corresponding tn ⊤ n subsets and k-subsets with their complements, we can identify (Wt,k ) with Wn−k,n−t . By Theorem 30(ii) with m = 2, when n − k ≤ n − t ≤ k, there is a vector ≡ 1 (mod 2) n in the row module of Wtk if and only if



(n − t) − i (n − k) − i



≡ 0 (mod 2)

implies



n−i (n − k) − i



≡ 0 (mod 2)

(25)

for i = 0, 1, . . . , n − k.   Lucas’ Lemma tells us that a binomial number ab is odd if and only if every power of 2 that appears in the base 2 representation of b also appears in that of a. If n = k + 2e where 2e is the least power of 2 thatappears 2 representation not in  t but k+2  in  the base  n of e e  ) n−t the base 2 representation of k, then n−k is = = k−(t−2 is even but e e 2 2 n−k n odd. That is, the implication (25) is false for i = 0, and hence 1 is not in row2 (Wtk ). Further analysis (we omit the details) shows that the implication (25) holds for all i when n < k + 2e .  If H is any t-uniform hypergraph and p a prime that divides the number of edges of H, we let R(H; p) denote the least integer n so that for any coloring of the edges of the (t) (t) t-uniform complete hypergraph Kn with Fp , there exists a subhypergraph H ′ of Kn ′ that is isomorphic to H and such that the sum of the colors on the edges of H is 0 in (t) Fp . So R(t, k; p) = R(Kk ; p). The problem of determining R(H; p) was introduced by Alon and Caro in [1], and in that paper they prove that for any graph G with k vertices and an even number of edges, R(G; 2) ≤ k + 2. This was generalized in [33] to Theorem 35 below. The number R(H; p) is the smallest integer n so that the p-ary code rowp (Nt ) does not contain a row vector all of whose coordinates are nonzero. Theorem 35 For any t-uniform hypergraph H on k vertices with an even number of edges, R(H; 2) ≤ k + t. Proof. >From Equation (24), when n ≥ k + t, Ett Nt = Dt Ft where Ft is rowunimodular. The first row of Ett is the vector of all 1’s, of length vt , and so the first row of Ett Nt is the constant vector of all g0 ’s, where g0 is the number of edges of H. The first entry of D is g0 , so the top row of Ft is the vector of all 1’s. The matrices Nt and Dt Ft have the same row module. A basis for rowp (Nt ) will consist of the rows of Ft that correspond to diagonal enties of D that are not divisible by p. If p divides g0 , the vector of all ones is not included, and it, of course, is not a linear combination of the other rows  of Ft . That is, (1, 1, . . . , 1) ∈ rowp (Ntn ) for n ≥ k + t. This can be improved to R(H; 2) ≤ k + t − 1 unless H is a complete t-uniform hypergraph; see [34]. Y. Caro has determined R(G; 2) for all simple graphs G.

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Theorem 36 (Caro [5]) Let G be a simple graph with k vertices and an even number of edges. Then R(G; 2) = k + 2 if G is complete, R(G; 2) = k + 1 if G is the union of two complete graphs or a non-complete graph with all vertices of odd degree, and R(G; 2) = k otherwise.

15. Diagonal forms for primitive hypergraphs and simple graphs In this section, we briefly state some recent joint results with Tony W. H. Wong. Proofs of the assertions and theorems in this section will appear in [34]. Let h be the characteristic t-vector of a t-uniform multihypergraph H. We say that H is primitive when the GCD of g, h over all integer t-vectors g in the null space of Wt−1,t is equal to 1. This concept of primitivity of hypergraphs appears implicitly in earlier work, e.g. [32]. Integer vectors in the null space of Wtk are called null designs or trades. A survey and comparison of explicit constructions of Z-bases for the null space of Wtk may be found in [15]. As an illustration, a Z-spanning set for the null space of W12 based on a v-set X is provided by 2-vectors g of the following form: Choose four points a, b, c, d in X and let g({a, b}) = g({c, d}) = 1,

g({a, d}) = g({b, c}) = −1,

(26)

and g({x, y}) = 0 otherwise. (This produces no vectors when v ≤ 3, but the null space v of W12 contains only the zero-vector in those cases.) That these vectors are orthogonal to the rows of W12 is easy, since the values of g on the pairs containing a vertex x ∈ X are either all zeros or all zeros except for a single +1 and a single −1. That the Z-span of these vectors is all of the null space of W12 is left to the reader to consider. This fact is known in many contexts. A form of it appears in [29], and all Z-bases for the null space of W12 discussed in [15] consist of vectors of this type. One must understand that primitivity of a hypergraph is affected by deleting or in(t) troducing isolated vertices. A complete t-uniform hypergraph Kk on k vertices is not primitive if t ≥ 1, since its characteristic t-vector h is constant and in the row module of (t) Wt−1,t so that the GCD of g, h is 0. But a hypergraph that is the union of Kk and at least t isolated vertices (and so with v ≥ k + t vertices) is primitive. This is related to the hypothesis v ≥ k + t in Theorem 32. A simple hypergraph with t − 1 isolated vertices is primitive unless the subhypergraph induced by the non-isolated vertices is complete; see [34]. By the shadow of a t-uniform multihypergraph H, we mean the (t − 1)-uniform multihypergraph H ′ where the multiplicity of a (t − 1)-subset S as an edge of H ′ is the sum of the multiplicities, as edges of H, of the t-subsets T that contain S. Or, in another terminology, if h is the characteristic t-vector of H, then the characteristic (t−1)vector of H ′ is Wt−1,t h. Similarly, the j-th shadow H (j) of H is the (t − j)-uniform multihypergraph with characteristic (t − j)-vector Wt−j,t h. Theorem 37 If a t-uniform multihypergraph H and all of its shadows H (j) , j = 1, 2, . . . , t, are primitive or multiples of primitive hypergraphs, then the diagonal entries of one diagonal form for Nt (H) are given by (23) where, as above, gi is the GCD of all entries of Wit Nt .

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This implies Theorem 32. When G is a simple graph and h its characteristic vector, the inner product g, h depends on the subgraph of G induced by {a, b, c, d} and is one of −2, −1, 0, 1, 2. For example, if a, b, c are the vertices of a triangle in G and d is adjacent to none of them, then for g as in (26), g, h = 0; but if d is adjacent to one of a or c, then g, h = ±1. Theorem 38 A simple graph G with at least four vertices is primitive unless G is isomorphic to a complete graph, an edgeless graph, a complete bipartite graph, or a disjoint union of two complete graphs. Even if a graph G is primitive, its shadow G′ may not be: if G is regular of positive degree, then G′ is a 1-uniform multihypergraph in which the multiplicity of every point is the same (i.e. its characteristic 1-vector is constant) and G′ is not primitive. Still, it is possible to give a unified description of the diagonal form of N2 (G) for primitive simple graphs. Theorem 39 Let G be a primitive simple graph with m ≥ 1 edges and degrees δ1 , δ2 , . . . , δn . Let h denote the GCD of the degrees δi and m; let g denote the GCD of all differences δi − δj , i, j = 1, 2, . . . , n. Then the invariant factors of N2 (G) are n

(1)( 2 )−n ,

(h)1 ,

(g)n−2 ,

(mg/h)1 .

(There need not be four distinct invariant factors. In particular, if g = 0, i.e. if G is regular, then mg/h = 0 and the last two factors combine to (0)n−1 .) The few nonprimitive simple graphs may be considered separately. We state one case here. Theorem 40 Let G be the complete bipartite graph Kr,n−r , where 2 ≤ r ≤ n − 2. Define m, g, and h as in the statement of Theorem 39, so in this case m = r(n − r),

g = n − 2r,

h = GCD{r, n − r}.

Then the diagonal entries of one diagonal form for N2 (G, n) are (1)n−2 ,

n

(2)( 2 )−2n+2 ,

(h)1 ,

(2g)n−2 ,

(mg/h)1 .

In the case r = 2, the matrix N2 is square; it is the adjacency matrix of the line graph of the complete graph Kn as mentioned in Section 3, and Theorem 40 contains the result in Brouwer and Van Eijl [4] stated in that section. Theorem 36 can be read as describing those graphs G for which (1, 1, . . . , 1) is in the binary code generated by N2 (G). Here is an extension of that result to odd primes p. Theorem 41 Let p be an odd prime and G a simple graph on n vertices. Define m, g, and h as in the statement of Theorem 39 and assume p divides m. Then (1, 1, . . . , 1) is in rowp (N2 (G)) if and only if (i) G is primitive, p|g, but p | h, (ii) G = K1,n−1 , (iii) G is the disjoint union of K1 and Kn−1 and p | n − 2, or (iv) G is the disjoint union of Kr and Kn−r for some r, 2 ≤ r ≤ n − 2, where also p|g, but p | h.

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Acknowledgements Supported in part by NSF Grant DMS-0555755. The author wishes to acknowledge helpful conversations with Tony W. H. Wong during the preparation of this manuscript.

References [1] N. Alon and Y. Caro, On three zero-sum Ramsey-type problems, J. Graph Th. 17 (1993), 177–192. [2] A. Blokhuis, A.; A. R. Calderbank, Quasi-symmetric designs and the Smith normal form, Des. Codes Cryptogr. 2 (1992), 189–206. [3] E. F. Assmus, Jr., and J. D. Key, Designs and their Codes, Cambridge Tracts in Mathematics 103, 1992. [4] A. E. Brouwer and C. A. van Eijl, On the p-rank of the adjacency matrices of strongly regular graphs, J. Alg. Combinatorics 1 (1992), 329–346. [5] Y. Caro, A complete characterization of the zero-sum (mod 2) Ramsey Numbers, J. Combinat. Thy. Ser. A 68 (1994), 205–211. [6] Y. Caro, Binomial coefficients and zero-sum Ramsey numbers, J. Combinatorial Th., Series A 80 (1997), 367–373. [7] David B. Chandler and Qing Xiang, The invariant factors of some cyclic difference sets, J. Combinatorial Th., Ser. A 101 131–146. [8] Z. Deretsky, On the symmetry of the Smith normal form for (v, k, λ) designs, Linear and Multilinear Algebra 14 (1983), no. 2, 187–193. [9] Anna Draganova, Yukiyasu Mutoh, and Richard M. Wilson, More on decompositions of edge-colored complete graphs, Discrete Mathematics 308 (2008), 2926–2943 [10] P. Frankl, Intersection theorems and mod p rank of inclusion matrices, J. Combinatorial Theory, Ser. A 54 (1990), 85–94. [11] R. L. Graham, S.-Y. R. Li, and W.-C. W. Li, On the structure of t-designs, SIAM J. Alg. Disc. Meth. 1 (1980), 8–14. [12] J. E. Graver and W. B. Jurkat, The module structure of integral designs, J. Combinatorial Theory 15 (1973), 75–90. [13] Marshall Hall, Jr. Combinatorial Theory (Reprint of the 1986 second edition), John Wiley & Sons, Inc., New York, 1998. [14] G. B. Khosrovshahi and S. Ajoodani-Namini, A new basis for trades, SIAM J. Discrete Math. 3 (1990), 364-372. [15] G. B. Khosrovshahi and Ch. Maysoori, On the bases for trades, Linear Algebra and its Appl., 226–228 (1995), 731–748. [16] G. B. Khosrovshahi; B. Tayfeh-Rezaie, A New proof of a classical theorem in design theory, J. Combinatorial Th., Series A 93 (2001), 391–396. [17] C. W. H. Lam, The search for a finite projective plane of order 10, Amer. Math. Monthly 98 (1991), 305–318. ˙ [18] Esther Lamken and Richard M. Wilson, Decompositions of edge-colored complete graphs, JCombin. Theory Ser. A 89 (2000), 149–200. [19] Eric S. Lander, Symmetric designs: an algebraic approach. London Mathematical Society Lecture Note Series 74, Cambridge University Press, Cambridge, 1983. xii+306 pp. [20] J. H. van Lint and R. M. Wilson, A Course in Combinatorics. Second edition. Cambridge University Press, Cambridge, 2001. [21] Eric G. Moorehouse, Projective Planes of Small Order, http://www.uwyo.edu/moorhouse/pub/planes/ (accessed 13 July, 2010). [22] Morris Newman, Invariant factors of combinatorial matrices, Israel J. Math. 10 (1971), 126–130. [23] Morris Newman, Integer Matrices, Academic Press, New york, 1972. [24] Morris Newman, The Smith normal form. Proceedings of the Fifth Conference of the International Linear Algebra Society (Atlanta, GA, 1995), Linear Algebra Appl. 254 (1997), 367–381. [25] Neil J. A. Sloane, A Library of Hadamard Matrices, http://www2.research.att.com/∼njas/hadamard/ (accessed 20 June 2010).

436 [26] [27] [28] [29]

[30] [31] [32] [33] [34]

R.M. Wilson / Codes and Modules Associated with Designs and t-uniform Hypergraphs H. J. S. Smith, On systems of linear indeterminate equations and congruences, Philos. Trans. Roy. Soc. London Ser. A 151 (1861), 293–326. R. C. Thompson, An inequality for invariant factors, Proceedings of the American Mathematical Society 86 (1982), 9–11. R. M. Wilson, The necessary conditions for t-designs are sufficient for something, Utilitas Mathematics 4 (1973), 207–217. R. M. Wilson, Decompositions of complete graphs into subgraphs isomorphic to a given graph, in: Proc. Fifth British Combinatorial Conference (C. St. J. A. Nash-Williams and J. Sheehan, eds.), Congressus Numerantium XV, Utilitas Mathematica Publ. (1975), 647–659. R. M. Wilson, A diagonal form for the incidence matrices of t-subsets vs. k-subsets, Europ. J. Combinatorics 11 (1990), 609–615. Richard M. Wilson, On set systems with restricted intersections modulo p and p-ary t-designs. Discrete Math. 309 (2009), 606–612. Richard M. Wilson, Signed hypergraph designs and diagonal forms for some incidence matrices. Des. Codes Cryptogr. 17 (1999) 289–297. Richard M. Wilson, Some applications of incidence matrices of t-subsets and hypergraphs, Proceedings of the Third Shanghai Conference on Combinatorics, Discrete Math., to appear. Richard M. Wilson and Tony W. H. Wong, Diagonal forms for incidence matrices arising from t-uniform hypergraphs, to appear.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved. doi:10.3233/978-1-60750-663-8-437

437

Finite geometry designs, codes, and Hamada’s conjecture Vladimir D. Tonchev Michigan Technological University, Houghton, Michigan, USA Abstract. The coding-theoretical interest in combinatorial designs defined by subspaces of a finite geometry was motivated in the 1960’s by their use for the construction of majority-logic decodable codes. In 1973, Hamada computed the ranks of the incidence matrices of finite geometry designs over the underlying finite field and made the conjecture that geometric designs have minimum rank among all designs with the given parameters. In all proved cases of the conjecture, the geometric designs not only have minimum rank, but are also the unique (up to isomorphism) designs of minimum rank. Until recently, only a handful of non-geometric designs were known that share the same rank with geometric designs. This paper discusses some recently discovered infinite families of non-geometric designs that have the same parameters and the same rank as certain geometric designs. Keywords. combinatorial design, incidence matrix, p-rank, finite geometry, majority-logic decoding, Hamada’s conjecture.

1. Introduction

Combinatorial designs have numerous applications to communications, information security and coding theory [5]. A large class of designs with diverse applications are the geometric designs defined by the subspaces of a given dimension of a finite affine or projective geometry. The coding-theoretical interest in geometric designs is motivated by their use for the construction of error-correcting codes with majority-logic decoding [32], [38]. The dimension of a code over GF (q) based on a combinatorial design is determined by the q-rank of its incidence matrix. In 1973, Hamada [12] computed the qranks of all geometric designs and made the conjecture that a design arising from a finite geometry over GF (q) has minimum q-rank among all designs with the given parameters. Hamada’s conjecture is discussed in Section 2. The fundamental questions concerning Hamada’s conjecture are whether all geometric designs have minimum q-rank, and whether the geometric designs are the only designs of minimum q-rank. In all proved cases of the conjecture, the geometric designs are characterized as the unique designs of minimum q-rank. Hamada’s conjecture is important for several reasons. It indicates that the geometric designs are the best choice for the construction of majority-logic decodable codes. The conjecture provides a computationally simple characterization of geometric designs, and implies the famous conjecture that any projective plane of prime order is Desarguesian. For more than 30 years, there have been only three known parameters sets for which a non-geometric design was known that had the same q-rank as a geometric design.

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The main topic of this paper are the recently discovered first infinite classes of nongeometric designs which have the same parameters and the same p-rank as the geometric design P Gd (2d, p) defined by the d-subspaces of the 2d-dimensional projective space P G(2d, p) over GF (p), where d ≥ 2 and p ≥ 2 is an arbitrary prime [20], as well as a class of 3-designs having the same 2-rank as the affine geometry designs AGd+1 (2d + 1, 2), d ≥ 2, having as blocks the (d + 1)-subspaces of the (2d + 1)dimensional affine space AG(2d+1, 2) over GF (2) [3]. These results indicate that not all geometric designs are characterized as the unique designs by their q-rank. The question whether the geometric designs are indeed of minimum q-rank is still widely open, with the exception of the few cases (discussed in Section 3) in which Hamada’s conjecture has been proved. A revised version of Hamada’s conjecture is considered in Section 4 that uses generalized incidence matrices with entries from the underlying field instead of (0, 1)incidence matrices.

2. Incidence matrices and Hamada’s conjecture A combinatorial t-(v, k, λ) design is a pair D = {X, B} of a finite set X of v points, and a collection B of b k-subsets of X called blocks, with the property that every t points are contained in exactly λ blocks [2], [5]. The incidence matrix of a design D is a b by v (0, 1)-matrix with rows indexed by the blocks and columns indexed by the points, where an entry is equal to 1 if the corresponding block and point are incident, and 0 otherwise. Two designs are isomorphic if there is a bijection between their point sets that maps the blocks of the first design to blocks of the second design. An automorphism of a design is a permutation of the points that preserves the collection of blocks. The q-rank of a design D, or rankq (D), is the rank of the incidence matrix of D over GF (q). Equivalently, the q-rank of a design is the dimension of the linear space over GF (q), or the linear q-ary code spanned by the rows of its incidence matrix. The interest in designs of low q-rank has been motivated by one of the early applications of combinatorial designs to coding theory, namely, for the construction of majoritylogic decodable codes (Rudolph [32]). A linear code C of length n whose dual code C ⊥ contains the blocks of a 2-(n, w, λ) design among the supports of its codewords of weight w, can correct up to (r + λ − 1)/(2λ) errors (where r = λ(n − 1)/(w − 1)) by majority-logic decoding, and possibly even a higher number of errors if C ⊥ supports t-designs with t > 2 (Rahman and Blake [31]). In particular, a linear code having as a parity check matrix the incidence matrix of a t-(v, k, λ) design with t ≥ 2 admits majority-logic decoding. However, since the total number of blocks of a t-design with t ≥ 2 and v > k > 0 is greater than or equal to the number of points (by the Fisher inequality [2]), the incidence matrix can be of full rank v, in which case the resulting code is trivial, consisting of the zero vector only. Therefore, for this purpose, it is important to choose a t-(v, k, λ) design of minimum rank (over the considered finite field) among all nonisomorphic designs having the given parameters. Most of the known majority-logic decodable codes are based on designs arising from finite geometries, a notable class of such codes being the Reed-Muller codes [1]. We refer to any design having as points and blocks the points and subspaces of a given dimension of a finite affine or projective geometry as a geometric design. We denote by P Gd (m, q)

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(resp. AGd (m, q)) the geometric design having as blocks the d-dimensional subspaces of the m-dimensional projective space P G(m, q) (resp. the m-dimensional affine space AG(m, q)) over GF (q). A projective geometry design P Gd (m, q) is a 2-(v, k, λ) design with parameters v= where

2 modified by replacing the q-rank of the incidence matrix with the q-dimension of the design over GF (q) may be true in general. A similar result was proved in [36] for the complete designs having as blocks all subsets of a given size. It is known that the minimum weight vectors in any MDS code, that is, a code which is optimal with respect to Singleton bound, support a complete design. Theorem 4.5. [36]. (i) The q-dimension dq of the complete design D = D(n, w) on a set X = {1, 2, . . . , n} having all w-subsets of X as blocks, is greater than or equal to n − w + 1. (ii) The equality qq = n − w + 1 holds if and only if there exists an [n, n − w + 1, w] MDS code over GF (q).

5. Non-geometric designs having the same p-rank as geometric designs There are no known examples of designs that have the same parameters, but smaller prank than a geometric design. However, there are examples of designs that violate the

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"only-if" part of the following stronger version of Hamada’s conjecture [1], which holds true in all proved cases of the conjecture: If D is a design with the same parameters as a design G having as blocks the ddimensional subspaces of AG(m, q) or P G(m, q), then rankq (D) ≥ rankq (G),

(4)

with equality rankq (D) = rankq (G) if and only if D is isomorphic to G. Until recently, there were only three known parameter sets of geometric designs for which there exist non-geometric designs with the same parameters and the same p-rank as their geometric counterparts, thus, demonstrating that the “only if" part of Hamada’s conjecture (4) is not true in general. These parameter sets are: 2-(31, 7, 7), 3-(32, 8, 7), [39] and 2-(64, 16, 5) [14]. 5.1. Quasi-symmetric 2-(31, 7, 7) designs and self-orthogonal 3-(32, 8, 7) designs A design having only two distinct block intersection numbers is called a quasi-symmetric design [33]. In [39], the the classification of binary doubly-even self-dual codes of length 32 was used to prove that up to isomorphism there are exactly five quasi-symmetric 2(31, 7, 7) designs, all having 2-rank equal to 16. One of the five quasi-symmetric designs is the geometric design P G2 (4, 2), while the remaining four are non-geometric designs that violate the “only if” part of Hamada’s conjecture. Two of the five quasi-symmetric 2-(31, 7, 7) designs, P G2 (4, 2) and the design supported by the binary quadratic-residue code of length 31, are cyclic (that is, have an automorphism of order 31). Goethals and Delsarte mentioned in their paper [9] that the two binary cyclic [31, 16, 7] codes, the quadratic-residue code and the punctured second order Reed-Muller code, support 2(31, 7, 7) designs. Each of the five quasi-symmetric 2-(31, 7, 7) designs extends to a 3-(32, 8, 7) design which is supported by the extended binary [32, 16] code. The resulting 3-(32, 8, 7) designs are self-orthogonal: their block intersection numbers are all even. It was proved in [39] that there exist exactly five non-isomorphic 3-(32, 8, 7) designs with even block intersection numbers, all having 2-rank 16. One of these five designs is the geometric design AG3 (5, 2), having as blocks the 3-subspaces of the binary affine geometry AG(5, 2), while the remaining four designs provide counter-examples to the strong form of Hamada’s conjecture in the affine case. 5.2. Non-geometric 2-(64, 16, 5) designs of 2-rank 16 In 2005, Harada, Lam and Tonchev [14] found two non-geometric 2-(64, 16, 5) designs having the same 2-rank as the classical geometric design AG2 (3, 4) having as blocks the planes of the 3-dimensional affine space AG(3, 4) over the field of order 4. The non-geometric 2-(64, 16, 5) designs found in [14] are the only known counter-examples to Hamada’s conjecture that correspond to geometric designs with classical parameters, i.e., having as blocks the hyperplanes of the corresponding finite geometry. These are also the only presently known counter-examples to Hamada’s conjecture over a field of non-prime order. The incidence vectors of the blocks of these designs were found as minimum weight vectors in binary linear codes spanned by the 64 by 64 incidence matrices of (4, 4)-nets.

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A symmetric (µ, q)-net is a 1-(µq 2 , µq, µq) design D such that both D and its dual design D∗ are affine resolvable [2]. The µq 2 points of a (µ, q)-net can be partitioned into µq disjoint parallel classes, each containing q points, so that any two points that belong to the same class do not occur together in any block, while any two points that belong to different classes occur together in exactly µ blocks. A symmetric (µ, q) net is class-regular if it admits a group of automorphisms G of order q (called group of bitranslations) that acts transitively (and hence regularly) on every point and block parallel class. Let n ≥ 2, and let q be a prime power. The classical (q n−2 , q) in AG(n, q) net is defined follows. The hyperplanes in AG(n, q) form an affine 2-design D with parameters (5): v = q n , k = q n−1 , λ =

q n−1 − 1 qn − 1 , r= q−1 q − 1.

(5)

Let P be a class of q n−1 parallel lines in AG(n, q), that is, P consists of a given onedimensional vector subspace L of GF (q)n and its cosets. Every line is contained in q n−1 − 1 (q n − q)(q n − q 2 ) . . . (q n − q n−2 ) = n−1 2 n−1 n−2 − q)(q − q ) . . . (q −q ) q−1

(q n−1

hyperplanes. Deleting from D the hyperplanes that contain lines from P leaves an incidence structure N with q n points being a 1-design such that every point is contained in exactly r′ = r −

q n−1 − 1 q n − 1 q n−1 − 1 = − = q n−1 q−1 q−1 q−1

blocks. Thus, N is an affine 1-(q n , q n−1 , q n−1 ) design and a symmetric (q n−2 , q)-net. Blocks of N are the hyperplanes that meet every line of P in at most one point. The group of translations of the one-dimensional vector subspace L is an elementary Abelian group G of order q that acts regularly on each line from P, and on each of the q n−1 parallel classes of blocks of N . Thus, G is a group of bitranslations of N , and N is a class-regular symmetric net. In [14], the symmetric class-regular (4, 4)-nets having a group of bitranslations G of order four were enumerated up to isomorphism. There are 226 nonisomorphic nets with G∼ = Z2 × Z2 , and and 13 nets with G ∼ = Z4 . Since the incidence vectors of the blocks of a symmetric (4, 4)-net are vectors of weight 16, and every two blocks are either disjoint or share 4 points, the incidence matrix of any such net spans a binary self-orthogonal code of length 64. The 2-ranks of the 226 class-regular (4, 4)-nets with a group of bitranslations Z2 × Z2 range from 16 to 25. The classical net in AG(3, 4) is one of the seven nets of 2-rank 16. Three of the seven codes of dimension 16 spanned by the incidence matrices of the nets, including the classical one, have the following Hamming weight enumerator: W (y) = 1 + 84y 16 + 3360y 24 + 17920y 28 + 22806y 32 + · · · + y 64 . In each of the three codes, the 84 vectors of weight 16 form the block by point incidence matrix of an affine 2-(64, 16, 5) design of 2-rank 16. The design in the code of the classical net is the geometric design AG2 (3, 4) having as blocks the planes in AG(3, 4).

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The 2-(64, 16, 5) designs supported by the other two codes are not only new counterexamples to Hamada’s conjecture, but also to the more restrictive Assmus conjecture that concerns geometric designs with classical parameters, having as blocks the hyperplanes in the corresponding finite geometry. 5.3. Line spreads in P G(2n + 1, 2) and Hamada’s conjecture Mavron McDonough, and the author [28] used a construction due to Rahilly [30] that relates affine designs with parallel classes of size four to symmetric Hadamard designs that possess a line spread, to analyze the non-geometric affine 2-(64, 16, 5) designs having the same 2-rank as the classical geometric design AG2 (3, 4). A line of a design through a pair of points x, y is the intersection of all blocks containing x and y. A line spread is a partition of all points of the design into disjoint lines. Let Γ be an affine 2-(16µ, 4µ, 13 (4µ−1)) design, where µ ≡ 1 (mod 4). We define a design Π as follows [30]. Let w be any point of Γ. The points of Π are all the points of Γ except w. The blocks of Π are defined as follows. Let C be a parallel class Γ, and let B0 be the block of C that contains w. For any B ∈ C with B = B0 , we define B ∪B0 −{w} to be a block of Π. It follows that Π is a symmetric 2-(16µ − 1, 8µ − 1, 4µ − 1) design and the three blocks B ∪ B0 − {w}, with B ∈ C and B = B0 , form a line in the dual Π∗ of Π, for any parallel class C. This construction is easily reversed to produce an affine 2-(16µ, 4µ, 13 (4µ − 1)) design from any symmetric Hadamard 2-(16µ − 1, 8µ − 1, 4µ − 1) design with a line spread. A computation of the Hadamard 2-(63, 31, 15) designs with a line spread obtained from the two exceptional affine 2-(64,16,5) designs of 2-rank 16 via Rahilly’s construction revealed the surprising fact that one of these two Hadamard 2-(63, 31, 15) designs is isomorphic to the geometric design P G4 (5, 2) [28]. Thus, certain line spreads of P G(2n + 1, 2) may be related to affine 2-(4n+1 , 4n , (4n − 1)/3) designs having the same 2-rank as the geometric affine design over GF (4) with these parameters. However, to find more such examples, one has to go to higher dimensions. All line spreads of P G(5, 2) were enumerated by Mateva and Topalova [27]. It turned out [27] that up to isomorphism, there are only two 2-(64, 16, 5) designs of 2-rank 16 that can be obtained via Rahilly’s construction from line spreads in P G(5, 2): the geometric design AG2 (3, 4), and one of the non-geometric 2-(64, 16, 5) designs found in [14].

6. Designs from polarieties in P G(2m − 1, q) The first infinite class of non-geometric designs that have the same p-rank as geometric designs was found recently by Jungnickel and the author [20]. This result motivates the search for an appropriate revision of Hamada’s conjecture and finding the range of dimensions, field characteristic, and field orders for which the conjecture is valid. A crucial tool in the construction of the new designs found in [20] were polarities in projective geometry. A correlation of a finite geometry G is a permutation α of the subspaces of G which inverts inclusion, i.e., S ⊆ T implies S α ⊇ T α for all subspaces S, T of G [7, p. 41]. A polarity is a correlation of order 2. A correlation α is a polarity if and only if S ⊆ T α implies S α ⊇ T for all subspaces S, T of G [7], [15].

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It was proved in [20] that every polarity of P G(2m − 1, q), where m ≥ 2, and q is an arbitrary prime power, gives rise to a design with the same parameters and the same block intersection numbers as, but not isomorphic to the design P Gm (2m, q) of points and m-subspaces of the projective space P G(2m, q). The smallest case, m = 2, yields a new infinite family of quasi-symmetric designs with parameters v=

q5 − 1 q3 − 1 q3 − 1 , k= , λ= , q−1 q−1 q−1

and block intersection numbers 1 and q + 1. The designs obtained via polarities share many properties with the geometric designs P Gm (2m, q). In particular, there is always a set H of q 2m−1 + . . . + q + 1 points on which the blocks induce an isomorphic copy of P G(2m−1, q), while a copy of an affine space AG(2m, q) is induced on the complementary set A of the remaining q 2m points. To prove that the new designs are not isomorphic to the geometric design P Gm (2m, q), the sizes of the lines in these designs were computed. In any of the new designs, the lines through two points of H or two points of A still have the natural geometric size, that is, q + 1 or q, respectively, whereas any point of H and a point of A determine a line of size 2. The construction of the new designs was suggested by a careful examination of properties of the five quasi-symmetric 2-(31, 7, 7) designs [39], and observing that one of the designs, having its points partitioned into two orbits of length 15 and 16 under its full automorphism group, shares the following property with the geometric design P G2 (4, 2): the restriction on the orbit of 15 points resembled a hyperplane in P G(4, 2). It turned out that this particular design can be obtained from P G2 (4, 2) via a permutation of the lines in a subspace P G(3, 2) defined by a polarity, and that observation led to the general construction. It was proved in [20] that the q-rank of a design obtained via a polarity is bounded by the q-rank of P Gm (2m, q) from below, and by ((q 2m+1 − 1)/(q − 1) + 1)/2 from above for arbitrary prime power q and any m ≥ 2. In the special case when q = p is a prime, it was proved that the new non-geometric designs have the same p-rank as the geometric design P Gm (2m, p), hence these designs provide an infinite class of counter-examples to the strong form of Hamada’s conjecture. An essential part of the the proof that the new non-geometric designs obtained via a polarity from P Gm (2m, p) when p is a prime have the same p-rank as the geometric design P Gm (2m, p), was the derivation of the following closed formula for the p-rank rp (m) of P Gm (2m, p): rp (m) =

1 p2m+1 − 1 ( + 1). 2 p−1

(6)

The simplest previously known version of Hamada’s formula for rp (m), found by Hirschfeld and Shaw [16], (see also [1, Theorem 5.8.1]) looks as follows: rp (m) =



m−1 (m − i)(p − 1) − 1 m + (m − i)p p2m+1 − 1  . (7) (−1)i − 2m − i i p−1 i=0

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It is worth noting that the equality between the two formulas (6) and (7) for rp (m) is an identity which holds true for arbitrary positive integer values of p [17], [25]. It is perhaps a little surprising that inequivalent polarities give rise to isomorphic designs via the polarity construction [20]. Computing the p-rank of designs obtained via polarities over fields of prime power order q = ps , s ≥ 2, is an interesting open problem. In the smallest case, q = 22 , the 2-rank of the geometric design P G2 (4, 4) is 146, while the 2-rank of the corresponding non-geometric design obtained from P G2 (4, 4) via a polarity, is 154 [20]. The construction based on polarities was extended in [3] to produce designs having the same parameters, intersection numbers, and 2-rank as the geometric design AGd+1 (2d + 1, 2) for any integer d ≥ 2. These designs generalize one of the four nongeometric 3-(32, 8, 7) designs of 2-rank 16 [39], and provide the only known infinite family of parameters for which affine geometry designs are not characterized by their rank. It was shown by Munemasa and the author [29] that the block graph of any nongeometric design obtained from P Gd (2d, q) via a polarity, is isomorphic to the twisted Grassmann graph discovered by van Dam and Koolen [6], which is a distance-regular graph with the same parameters as the Grassman graph Jq (2d + 1, d).

References [1] E.F. Assmus and J.D. Key, Designs and Their Codes, Cambridge University Press, Cambridge 1992. [2] T. Beth, D. Jungnickel, H. Lenz, Design Theory, Second Edition, Cambridge University Press, Cambridge 1999. [3] D. Clark, D. Jungnickel, and V.D. Tonchev, Affine geometry designs, polarities, and Hamada’s conjecture, J. Combin. Theory, Ser. A, to appear. [4] D. Clark, D. Jungnickel, and V.D. Tonchev, Exponential bounds on the number of designs with affine parameters, J. Combin. Designs, to appear. [5] C. J. Colbourn and J.F. Dinitz, eds., Handbook of Combinatorial Designs, Second Edition, CRC Press, Boca Raton, 2007. [6] E.R. van Dam and J.H. Koolen, A new family of distance-regular graphs with unbounded diameter, Invent. Math. 162 (2005), 189-193. [7] P. Dembowski, Finite Geometries, Springer, Berlin, 1968. [8] J. Doyen, X. Hubaut, M. Vandensavel, Ranks of incidence matrices of Steiner triple systems, Math. Z. 163 (1978), 251–259. [9] J.M. Goethals and P. Delsarte, On a class of majority logic decodable cyclic codes, IEEE Trans. Info. Theory 14 (1968), 182-188. [10] R.L. Graham and J. MacWilliams, On the number of information symbols in different difference-set cyclic codes, Bell Sys. Tech. J. 45 (1966), 1057-1070. [11] M. Hall, Jr., Combinatorial Theory, Second Edition, Wiley, New York, 1986. [12] N. Hamada, On the p-rank of the incidence matrix of a balanced or partially balanced incomplete block design and its application to error-correcting codes, Hiroshima Math. J. 3 (1973), 153-226. [13] N. Hamada and H. Ohmori, On the BIB design having the minimum p-rank, J. Combin. Theory A 18 (1975), 131-140. [14] M. Harada, C. Lam, and V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs, Codes, and Cryptography 34 (2005), 71-87. [15] J.W.P. Hirschfeld, Projective Geometries over Finite Fields, Second Edition, Oxford University Press, Oxford, 1998. [16] J. W. P. Hirschfeld and R. Shaw: Projective geometry codes over prime fields. In: Finite Fields: Theory, Application and Algorithms. Contemporary Math 168 (1994), pp. 151–163. Amer Math. Soc., Providence, R.I.

448 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

V.D. Tonchev / Finite Geometry Designs, Codes, and Hamada’s Conjecture J. L. W. V. Jensen: Sur une identité d’Abel et sur d’autres formules analogues, Acta Math. 26 (1902), 307-318. D. Jungnickel, The number of designs with classical parameters grows exponentially, Geom. Dedicata 16 (1984), 167–178. D. Jungnickel, Characterizing Geometric Designs, II, J. Combin. Theory, Ser. A, to appear. D. Jungnickel and V.D. Tonchev, Polarities, quasi-symmetric designs, and Hamada’s conjecture, Designs, Codes and Cryptography, 51 (2009), 131-140. D. Jungnickel and V.D. Tonchev, The number of designs with geometric parameters grows exponentially, Designs, Codes and Cryptography, 55 (2010), 131-140. W. M. Kantor: Automorphisms and isomorphisms of symmetric and affine designs. J. Algebraic Combin. 3 (1994), 307–338. C. Lam, S. Lam, V. Tonchev, Bounds on the number of affine, symmetric and Hadamard designs and matrices, J. Combin. Theory, Ser. A 92 (2000), 186–196. C. Lam and V. D. Tonchev: A new bound on the number of designs with classical affine parameters. Designs, Codes and Cryptography 27 (2002), 111–117. M. E. Larsen: Summa Summarum, CMS Treatises in Mathematics, Canadian Mathematical Society, Ottawa, ON; A K Peters, Ltd., Wellesley, MA (2007). F.J. MacWilliams and H.B. Mann, On the p-rank of the design matrix of a difference set, Information and Control 12 (1968), 474-488. Z. Mateva, S. Topalova, Line spreads of P G(5, 2), J. Combin. Designs 17 (2009), 90-102. V. C. Mavron, T.P. McDonough, and V.D. Tonchev, On affine designs and Hadamard designs with line spreads, Discrete Math. 308 (2008), 2742-2750. A. Munemasa and V.D. Tonchev, The twisted Grassmann graph is the block graph of a design, Innovations in Incidence Geometry, to appear. A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991), 291-303. M. Rahman and Ian F. Blake, Majority logic decoding using combinatorial designs, IEEE Trans. Info. Theory 21 (1975), 585-587. L.D. Rudolph, A class of majority-logic decodable codes, IEEE Trans. Info. Theory 13 (1967), 305-307. M. S. Shrikhande and S. S. Sane, “Quasi-Symmetric Designs”, LMS Lecture Note Ser. 164, Cambridge 1991. K.J.C. Smith, On the p-rank of the incidence matrix of points and hyperplanes in a finite projective geometry, J. Combin. Theory 7 (1969), 122-129. L. Teirlinck, On projective and affine hyperplanes, J. Combin. Theory Ser. A 28 (1980), 290-306. V.D. Tonchev, A note on MDS codes, n-arcs and complete designs, Designs, Codes and Cryptography 29 (2003) , 247-250. V.D. Tonchev, Linear Perfect Codes and a Characterization of the Classical Designs, Designs, Codes and Cryptography 17 (1999), 121-128. V.D. Tonchev, “Codes and Designs”, Chapter 15, Volume II, pp. 1229-1268 in: Handbook of Coding Theory, V.S. Pless and W.C. Huffman eds., North Holland, Amsterdam 1998. V.D. Tonchev, Quasi-symmetric 2-(31,7,7) designs and a revision of Hamada’s conjecture, J. Combin. Theory, Ser. A 42 (1986), 104-110.

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved.

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Subject Index AES 38 algebraic construction 326 authentication code 326 balanced array 326 Balanced generalized weighing matrix 363 binary codes 253 Block designs 253 Bush-type Hadamard matrices 312 codes 172, 202, 231 combinatorial design 437 combinatorial enumeration 27 combinatorial identities 27 comma-free code 326 Complex Hadamard matrix 312 conjugacy classes 202 correlation 136 covering array 99 design 172, 202, 231 difference sets 136 difference system of sets 326 discrete logarithm 1 distributing hash family 99 divisible designs 253 error-correcting code 278 experimental design 326 finite geometries 38, 172, 437 finite groups 1 frequency hopping sequence 326 Generalized Hadamard matrix 363 genetic programming 17 genus two curves 59 geometric construction 326 graphs 172 Griesmer bound 38 Hadamard matrices 253, 312 Hamada’s conjecture 437 heterogeneous hash family 99 hyperelliptic curve cryptography 59 hypergraphs 404 incidence matrices 404, 437 information security 136

interaction testing 99 intractability 1 Jacobsthal numbers 27 large set 285 majority-logic decoding 437 maximal subgroups 202 MDS codes 38 modular polynomials 59 moduli spaces 59 MOLS 312 MRHS 17 multi-structured design 326 Mutually suitable Latin squares 312 mutually unbiased weighing matrices 312 nested design 326 optical orthogonal code 326 perfect hash family 99 PET SNAKE 17 p-rank 404, 437 primitive simple group 231 PSL(2, p) 1 public key cryptosystems 1 quantum jump code 285 Quasi-derived design 363 Quasi-residual design 363 radar 136 Regular Hadamard matrix 363 representations and presentations of groups 1 rooted forest 17 rooted tree 17 row-column design 326 secret sharing 38 Seidel switching 253 self-dual 278 separating hash family 99 sequences 136 simple groups 202 Smith normal form 404 strongly regular graphs 231, 253 symmetric design 363

450

synchronization Tillich-Zémor hash function t-SEED t-uniform hypergraphs ultra-wideband

136 1 285 404 326

unbiased bases unbiased complex Hadamard matrices uniform random generation weighing matrix

312 312 17 312

Information Security, Coding Theory and Related Combinatorics D. Crnković and V. Tonchev (Eds.) IOS Press, 2011 © 2011 The authors and IOS Press. All rights reserved.

451

Author Index Alraqad, T.A. Arasu, K.T. Beshaj, L. Colbourn, C.J. Crnkovi, D. Fuji-Hara, R. Haemers, W.H. Ili, I. Jimbo, M. Key, J.D. Kharaghani, H. Klein, A. Lam, C. Magliveras, S.S.

363 136 59 99 231 326 253 1 285 172 312 38 278 1, 17, 27

Matheis, K. Miao, Y. Mikuli Crnkovi, V. Moori, J. Rodrigues, B.G. Shaska, T. Shiromoto, K. Shrikhande, M.S. Storme, L. Tonchev, V.D. van Trung, T. Wei, W. Wilson, R.M.

17 326 231 202 231 59 285 363 38 437 27 27 404

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