Graph Theory, Coding Theory and Block Designs 9780521207423, 0521207428

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London Mathematical Society Lecture Note Series Mi1§

Graph Theory Coding Theory and Block Designs '

P.J.CAMERON and J.H.VAN LINT

1C6.3

CAMBRIDGE UNIVERSITY PRESS

NUNC COGNOSCO EX PARTE

THOMAS J. BATA LIBRARY TRENT UNIVERSITY

LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Editor: PROFESSOR G. C. SHEPHARD, University of East Anglia This series publishes the records of lectures and seminars on advanced topics in mathematics held at universities throughout the world. For the most part, these are at postgraduate level either presenting new material or describing older material in a new way. Exceptionally, topics at the undergraduate level may be published if the treatment is sufficiently original. Prospective authors should contact the editor in the first instance. Already published in this series 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 17.

General cohomology theory and K-theory, PETER HILTON. Numerical ranges of operators on normed spaces and of elements of normed algebras, F. F. BONSALL and J. DUNCAN. Convex polytopes and the upper bound conjecture, P. MCMULLEN and G. C. SHEPHARD. Algebraic topology: A student's guide, J. F. ADAMS. Commutative algebra, J. T. KNIGHT. Finite groups of automorphisms, NORMAN BIGGS. Introduction to combinatory logic, J. R. HINDLEY, B. LERCHER and J. P. SELDIN. Integration and harmonic analysis on compact groups, R. E. EDWARDS. ' Elliptic functions and elliptic curves, PATRICK DU VAL. Numerical ranges II, F. F. BONSALL and J. DUNCAN. New developments in topology, G. SEGAL (ed. ). Proceedings of the Symposium in Complex Analysis Canterbury 1973, J. CLUNIE and W. K. HAYMAN (eds. ). Combinatorics, Proceedings of the British Combinatorial Con¬ ference 1973, T. P. MCDONOUGH and V. C. MAVRON (eds. ). Analytic theory of abelian varieties, H. P. F. SWINNERTONDYER. Introduction to topological groups, P. J. HIGGINS. Differential germs and catastrophes, THEODOR BRb’CKER and L. C. LANDER.

London Mathematical Society Lecture Note Series.

19

Graph Theory Coding Theory and Block Designs RJ.CAMERON & J.H.VAN LINT

Cambridge University Press Cambridge London - New York - Melbourne

Published by the Syndics of the Cambridge University Press Bentley House, 200 Euston Road, London NW1 2DB American Branch:

©

32 East 57th Street, New York, N. Y. 10022

Cambridge University Press 1975

Library of Congress Catalogue Card Number:

ISBN:

0 521 20742 8

Printed in Great Britain at the University Printing House, Cambridge (Euan Phillips, University Printer)

74-25645

Contents

Page Introduction

v

1.

A brief introduction to design theory

1

2.

Strongly regular graphs

11

3.

Quasi-symmetric designs

19

4.

Strongly regular graphs with no triangles

25

5.

Polarities of designs

33

6.

Extension of graphs

38

7.

Codes

44

8.

Cyclic codes

51

9.

Threshold decoding

57

10.

Reed-Muller codes,

59

11.

Self-orthogonal codes and designs

65

12.

Quadratic residue codes

71

13.

Symmetry codes over

81

14.

Nearly perfect binary codes and uniformly packed codes

87

15.

Association schemes

96

GF(3)

References

107

Index

113

Introduction

In the year 1973, Professor van Lint and Dr. Cameron visited Westfield College separately on various occasions to lecture at our Seminar on Combinatorial Algebra and Geometry.

As the subject matter

of their lectures overlapped in a most interesting way, the idea was conceived of organizing the notes already prepared by each of them into a coherent whole.

The present volume is the outcome.

The object of the lectures, and of these lecture notes, was to present to an audience already familiar with the theory of designs some of the connections and applications with other branches of mathematics. (For the present book, however, a short introductory chapter on designs has been included.) In particular, this has meant largely graphs and codes, and to both of these subjects as well some kind of introduction is ✓

given.

But the object has been to stress the connections of design theory

with these other areas, and so no attempt at a consistent coverage of graph theory and coding theory has been made. The perceptive reader may well detect a difference in style between various chapters, revealing the particular approaches of the two authors.

But we believe that the overall mathematical unity of the

lectures and of the notes will be evident, and very useful to students and research workers in these fields.

D. R. Hughes

v

Digitized by the Internet Archive in 2019 with funding from Kahle/Austin Foundation

https://archive.org/details/graphtheorycodinOOOOcame

1 ■ A brief introduction to design theory

These lectures were given to an audience of design theorists; for those outside this class, the introductory chapter describes some of the concepts of design theory and examples of designs to which we shall refer. A t-design with parameters (v, k, A) is a collection 2}

(or a t - (v, k, X) design)

of subsets (called blocks) of a set S of v points,

such that every member of 25 contains k points, and any set of t points is contained in exactly X members of 25.

Various conditions are usually

appended to this definition to exclude degenerate cases. S and 25 with A = 1

We assume that

are non-empty, and that v>k>t (so A> 0). is called a Steiner system.

A t-design

Alternatively, a t-design may be

defined to consist of a set of points and a set of blocks, with a relation called incidence between points and blocks, satisfying the appropriate conditions. Sometimes a t-design is defined so as to allow 'repeated blocks’, that is, 25 is a family rather than a set, and the same subset of S may occur more than once as a block.

(This is more natural with the 'rela¬

tion' definition; simply omit the condition that any k points are incident with at most one block. ) In these notes, we normally do not allow repeated blocks; where they are permitted, we shall say so.

There are 'non¬

trivial' t-designs with repeated blocks for every value of t; but the only known examples without repeated blocks with t > 5 are those in which every set of k points is a block.

The existence of non-trivial t-designs

with t > 5 is the most important unsolved problem in the area; even 5-designs are sufficiently rare that new constructions are interesting. Of course, for Steiner systems the question of repeated blocks does not arise.

Only two Steiner systems with t = 5, and two with t = 4 are

known; these will be described later.* * Several more such systems have recently been constructed by R. H. F. Denniston.

1

Remark.

A 0-design is simply a collection of k-element subsets

of a set. In a t-design, let A^ denote the number of blocks containing a given set of i points, with 0 < i < t.

Counting in two ways the number

of choices of t - i further points and a block containing all t distin¬ guished points, we obtain

(1.1)

V -

v?: i> = A, the matrix M M is non-singular, from which follows Fisher's inequality: (1. 5)

Theorem.

In a 2-design with k < v - 1, we have b^v,

Furthermore, if b = v, then MJ = JM; thus M commutes with (r - A)I + AJ, and so with ((r - A)I + AJ)M 1 = M^. So T MM = (r - A)I + AJ, from which we see that any two blocks have A common points. (1. 6)

Theorem.

In a 2-design with k < v - 1, the following are equiva¬

lent: (i)

b = v;

(ii)

r = k;

(iii)

any two blocks have A common points.

A 2-design satisfying the conditions of (1. 6) is called symmetric. Its dual is obtained by reversing the roles of points and blocks, identifying a point with the set of blocks containing it; the dual is a symmetric 2T design with the same parameters, having incidence matrix M . A polarity of a symmetric design D is a self-inverse isomorphism between 2) and its dual, that is, a one-to-one correspondence a between the points and blocks of 2)

such that, for any point p and block B, p e B

if and only if

A point p (resp. a block B) is absolute with

epff,

respect to the polarity a if p e pCT (resp.

Ba e B).

(1. 5) and (1. 6) follow from a more general result, which we will need in chapter 4.

3

(1.7)

Theorem.

In a 2-design, the number of blocks not disjoint from

a given block B is at least k(r-l)2/((k-l)(A-l) + (r-1)).

Equality holds

if and only if any block not disjoint from B meets it in a constant number of points; if this occurs, the constant is 1 + (k-l)(A-l)/(r-l). Proof.

Let B , ... , Bd be the blocks different from but not

disjoint from B, and x = | B n B. |.

Counting in two ways the number

of choices of one or two points of B and another block containing them, we obtain (with summations running from 1 to d): Ex. = k(r - 1), E x.(x. - 1) = k(k - 1)(A - 1). So E (x. - x)2 = dx2 - 2k(r - l)x + k((k - 1)(A - 1) + (r - 1)). This quadratic expression in x must be positive semi-definite, and can vanish only if all x^ are equal; the common value must then be 1 + (k - 1)(A - l)/(r - 1). y Remark.

Now (1. 5) follows from b-1 > k(r-l)2/((k-l)(A-l)+(r-l))

on applying (1. 2) and (1. 3).

Also, if b = v, then r =k, and

1 + (k-1) (A-1) /(r -1) = A. The Bruck-Ryser-Chowla theorem gives necessary conditions for the existence of symmetric designs with given parameter sets (v, k, A) satisfying (v - 1)A = k(k - 1). (1. 8)

Theorem.

Put n = k - A.

Suppose there exists a symmetric 2 - (v, k, A) design.

Then (i) if v is even then n is a square; (ii) if v is

odd, then the equation z

= nx

+ (-1)

Ay

has a solution in integers (x, y, z), not all zero. T The incidence equation M M = nl + AJ shows that the matrices I and nl + AJ are rationally cogredient; (1. 8) can then be verified by

4

aPPlying the Hasse-Minkowski theorem to them, though more elementary proofs are available.

The Hasse-Minkowski theorem guarantees the

existence of a rational matrix M satisfying the incidence equation when¬ ever the conditions of (1. 8) are satisfied; but this does not mean that a design exists, and indeed it is not known whether these conditions are sufficient for the existence of a design.

We shall have more to say about

the case (v, k, A) = (111, 11, 1) in chapter 11. A Hadamard matrix is an n x n matrix H with entries ±1 satisT T tying HH = H H = nl. (It is so called because its determinant attains a bound due to Hadamard.) Changing the signs of rows and columns leaves the defining property unaltered, so we may assume that all entries in the first row and column are +1.

If we then delete this row and column

and replace -1 by 0 throughout, we obtain a matrix M which (if n> 4) is the incidence matrix of a symmetric 2 - (n-1, f n-1, in-1) design. Such a design is called a Hadamard 2-design.

From a design with these

parameters, we can recover a Hadamard matrix by reversing the con¬ struction.

However, a Hadamard matrix can be modified by permuting

rows and columns, so froin 'equivalent' Hadamard matrices it is possible to obtain different Hadamard 2-designs. Let H be a Hadamard matrix with n > 4, in which every entry in the first row is +1.

Any row other than the first has In entries +1

and in entries -1, thus determining two sets of fn columns.

(This

partition is unaffected by changing the sign of the row.) If we take columns as points, and the sets determined in this way as blocks, we obtain a 3 - (n, in, i-n-1) design called a Hadamard 3-design.

Any design with

these parameters arises from a Hadamard matrix in this way. It should be pointed out that Hadamard matrices are very plentiful. Examples are known for many orders n divisible by 4 (the smallest un¬ settled case at present is n = 188), and for moderately small n there are many inequivalent matrices. A projective geometry over a (skew) field F is, loosely speaking, the collection of subspaces of a vector space of finite rank over points of the geometry are the subspaces of rank 1.

F.

The

A projective geo¬

metry is often regarded as a lattice, in which points are atoms and every

5

element is a join of atoms.

We shall identify a subspace with the set of

points it contains, regarded as a subset of the point set.

The dimension

of a subspace is one less than its vector-space rank (so points have dimension 0); the dimension of the geometry is that of the whole space. Lines and planes are subspaces of dimension 1 and 2 respectively; hyper¬ planes are subspaces of codimension 1.

Thus, familiar geometric state¬

ments hold: two points lie in a unique line, a non-incident point-line pair lies in a unique plane, etc. If F is finite, the subspaces of given positive dimension are the blocks of a 2-design.

The hyperplanes form a symmetric 2-design, which

we shall denote by PG(m, q), where m is the dimension and q = |f|; its parameters are ((qm+1 - l)/(q - 1), (qm - l)/(q - 1), (qm 1 - l)/(q-l)). These facts can be verified by counting arguments, or by using the transi¬ tivity properties of the general linear group.

Note that PG(m, 2) is a

Hadamard 2-design for all m ^ 2. A projective plane is a symmetric 2-design with A = 1.

This

agrees with the previous terminology, in that PG(2, q) is a projective plane; by analogy, the blocks of a projective plane are called lines.

We

will modify our notation and use PG(2, q) to represent any projective plane with k = q + 1; q is not even restricted to be a prime power (though it is in all known examples).

Thus PG(2, q) may denote several

different designs; however, the symbol PG(m, q) is unambiguous for m > 2.

This is motivated by various characterisations; we mention one

due to Veblen and Young [69]: (1. 9)

Theorem.

A collection of subsets (called Tines') of a finite set of

points is the set of lines of a projective geometry or projective plane if the following conditions hold: (i)

any line contains at least three points, and no line contains

every point; (ii)

any two points lie on a unique line;

(iii)

any three non-collinear points lie in a subset which, together

with the lines it contains, forms a projective plane. Projective planes can be axiomatised in a way which permits the extension of the definition (and of (1. 9)) to infinite planes; the require-

6

merits are that any two points lie on a unique line, any two lines have a unique common point, and there exist four points of which no three are collinear.

Projective planes over fields are called Desarguesian, since

they are characterised by the theorem of Desargues. An affine (or Euclidean) geometry of dimension m is the collection of cosets of subspaces of a vector space of rank m over a field F. Here, the geometric dimension of a coset is equal to the vector-space dimension of the underlying subspace; points are just vectors (or cosets of the zero subspace). points it contains.

Again, we identify a subspace with the set of

If F is finite, the subspaces of given positive dimen¬

sion form a 2-design.

If

| F | =2, then any line has two points (and any

set of two points is a line), while the subspaces of given dimension d > 1 form a 3-design.

We denote the design of points and hyperplanes by

AG(m, q), where q= |f|.

AG(m, 2) is a Hadamard 3-design.

An affine plane AG(2, q) can be defined to be a 2 - (q2, q, 1) design.

(It is also possible to give an axiomatic definition of an affine

plane, in terms of the concept of 'parallelism'. ) Again our notation is ambiguous.

The exact analogue of the Veblen-Young theorem has been

proved by Buekenhout [15] under the restriction that each line has at least four points; Hall [33] has given a counterexample with three points on any line.

Again, affine planes over fields are characterised by Desargues'

theorem. Finally, we note that an affine geometry can be obtained from the corresponding projective geometry by deleting a hyperplane (the 'hyper¬ plane at infinity') together with all of its subspaces, while the projective geometry can be recovered from the affine geometry.

The same is true

for projective and affine planes. We shall consider briefly in chapter 4 the affine symplectic geometry of dimension 4 over GF(2).

Here the vector space carries a

symplectic bilinear form, and the blocks of the design are the cosets of those subspaces of rank 2 which are totally isotropic with respect to the form. Given a t-design

33, the derived design 33^ with respect to a

point p is the (t-l)-design whose points are the points of 33 different from p, and whose blocks are the sets B- ip]

for each block B of

7

2) which contains p.

The residual design 2)P with respect to p has the

same point set as 2) , but its blocks are the blocks of 2) not containing p; it is also a (t-1)-design. A converse question, which is very important in design theory and permutation groups, is that of extendability: given a t-design, is it isomorphic to D is called an extension of the given design. or there may be more than one. find a suitable design 2)P. )

for some (t+l)-design 2)?

2)

An extension may not exist,

(To construct the extension, we must

By applying (1. 2) to the extension, we obtain

a simple necessary condition for extendability: (1. 10) Proposition.

If a t - (v, k, A) design with b blocks is extend-

able, then k + 1 divides b(v + 1). The 2-design PG(2, q) has v = q£ + q+ l = b, k = q + 1. Applying (1. 10), we obtain a result of Hughes [39]: (1. 11) Theorem.

If PG(2, q) is extendable, then q = 2, 4, or 10.

Much effort has been devoted to these projective planes and their extensions.

Further application of (1. 10) shows that PG(2, 2) and

PG(2, 10) can be extended at most once, and PG(2, 4) at most three times.

In fact, PG(2, 2) is unique, and has a unique extension, namely

AG(3, 2).

PG(2, 4) is also unique, and can be extended three times,

each successive extension being unique (up to isomorphism).

The exis¬

tence of PG(2, 10) is still undecided (we will discuss this further in chapter 11) and its extendability appears rather remote at present. Hughes showed further that there are only finitely many extendable symmetric 2 - (v, k, A) designs with any given value of A. est result in this direction is due to Cameron [17],

The strong¬

We will use it (and

indicate the proof) in chapter 4. (1. 12) Theorem.

If a symmetric

2 - (v, k, A) design 2)

then one of the following occurs:

8

(i)

21

is a Hadamard 2-design;

(ii)

v = (A + 2)(A(i) 2 + 4A + 2), k = A2 + 3A + 1;

(iii)

v = 111, k = 11, A = 1;

is extendable,

(iv)

V

= 495, k = 39, A = 3.

Regarding case (i) of this result, a Hadamard 2-design is uniquely extendable.

For it is not hard to show that, in a Hadamard 3-design,

the complement of a block is a block; then the unique extension of the Hadamard 2-design 2) is obtained by adding the 'extra point' to each block of 5)

and then defining the complement of each such block to be

also a block in the extension.

Apart from Hadamard designs, the only

known extendable symmetric 2-design is PG(2, 4)

(case (ii) with A = 1);

it is also the only twice-extendable symmetric 2-design. For affine planes, the situation is a little different, since the necessary condition (1. 10) is always satisfied.

An extension of an affine

plane (that is, a 3-(q"+l, q + 1, 1) design) is called an inversive plane or M'obius plane.

Many examples are known; the affine planes over finite

fields are all extendable, sometimes in more than one way.

However,

Dembowski [22], [23] has shown that an inversive plane with even q can be embedded in PG(3, q) in a natural way (so q is a power of 2).

Using

this and a further embedding technique in conjunction with (1. 10), Kantor [42] showed that if AG(2, cf) is twice extendable (with q > 2), then q = 3 or 13.

In fact, AG(2, 3) is three times extendable; the extensions

are 'embedded' in the corresponding extensions of PG(2, 4), as well as in the projective geometries over GF(3).

It is not known whether any

AG(2, 13) is twice extendable. The extensions of PG(2, 4) and AG(2, 3) are so important that we shall give a brief outline of their construction.

See also Witt [73],

[74], LOneberg [46], Todd [68], Jonsson [41], etc. Starting from PG(2, 4), the 5 - (24, 8, 1) design is constructed by adding three points p, q, r; the blocks containing all three of these points are the sets

{p, q, r }

U

L, where L is a line of PG(2, 4).

We

must specify the blocks which do not contain all three of these points, as subsets of PG(2, 4).

They turn out to be natural geometric objects:

hyperovals, subplanes PG(2, 2), and symmetric differences of pairs of lines.

Indeed, these are the only possible candidates; in this way the

uniqueness of the designs can be shown.

(Important for this is the fact,

proved by simple counting, that two blocks of a 5 - (24, 8, 1) design

9

have 0, 2 or 4 common points. ) Also, if U is a unital in PG(2, 4)

(p, q, r } u U

set of absolute points of a polarity of unitary type), then is the point set of a 5 - (12, 6, 1) design.

(the

Alternatively, the latter

design can be obtained by extending AG(2, 3) three times, identifying the added blocks with geometric objects in the plane as before. Another construction for the 5 - (12, 6, 1) design uses the fact that the symmetric group S

has an outer automorphism.

sets of six elements on which S

6

Taking two

acts in the two possible ways, the

blocks of the 5 - (12, 6, 1) design can be defined in terms of the permu¬ tations.

This method also gives a uniqueness proof.

continued: the automorphism group

The process can be

of the design itself has an outer

automorphism, and a similar construction produces the 5 - (24, 8, 1) design. It is possible to construct the 5-fold transitive Mathieu groups M

and M

directly, and to deduce properties of the designs from

them. The techniques-of coding theory can be used to construct the design in a purely algebraic way.

We shall describe this in chapters 11 and 12.

In higher dimensions, the situation is simpler.

PG(m, 2)

(with

m > 2) has an extension only if q = 2, when the Hadamard 3-design AG(m + 1, 2) is the unique extension; AG(m, q) is not extendable for m > 2. For further reading, see Dembowski [24], chapter 2 for designs, section 1. 4 for projective and affine geometries, chapters 3-5 for pro¬ jective planes, and chapter 6 for inversive planes.

Block designs are

also discussed in the books by Hall [33] and Ryser [56].

10

2 • Strongly regular graphs

The theory of designs concerns itself with questions about subsets of a set (or relations between two sets) possessing a high degree of sym¬ metry.

By contrast, the large and amorphous area called 'graph theory'

is mainly concerned with questions about 'general' relations on a set. This generality means that usually either the questions answered are too particular, or the results obtained are not powerful enough, to have useful consequences for design theory.

There are some places where the two

theories have interacted fruitfully; in the next five chapters, several of these areas will be considered.

The unifying theme is provided by a

class of graphs, the 'strongly regular graphs', introduced by Bose [11], whose definition reflects the symmetry inherent in t-designs.

First,

however, we shall mention am example of the kind of situation we shall not be discussing. A graph consists of a finite set of vertices together with a set of edges, where each edge is a subset of the vertex set of cardinality 2. (In classical terms, our graphs are undirected, without loops or multiple edges.) As with designs, there is an alternative definition: a graph con¬ sists of a finite set of vertices and a set of edges, with an 'incidence' relation between vertices and edges, with the properties that any edge is incident with two vertices and any two vertices with at most one edge. Still another definition: a graph consists of a finite set of vertices to¬ gether with a symmetric irreflexive binary relation (called adjacency) on the vertex set.

A graph is complete if every pair of vertices is adjacent,

and null if it has no edges at all.

The complement of the graph T is the

graph T whose edge set is the complement of the edge set of T (in the set of all 2-element subsets of the vertex set).

In any graph T,

will denote the set of vertices adjacent to the vertex p. S of the vertex set,

r|s

T(p)

Given a subset

denotes the graph with vertex set S whose

11

edges are those edges of r which join pairs of vertices of S. For example, let n be a natural number, and let V be the set of all n x n latin squares with entries

{l, .. . , n }.

Form a graph

Tn with vertex set V by saying that two latin squares are adjacent if and only if they are orthogonal.

Now it is well-known that there exists

a (projective or affine) plane of order n if and only if complete graph on n - 1 vertices as a subgraph.

contains the

Graph theory provides

us with lower bounds for the size of complete subgraphs of graphs; not surprisingly, these are far too weak to force the existence of a plane of given order except in trivial cases.

This simple approach seems unlikely

to produce any useful results for the theory of finite planes. If p is a vertex of a graph T, the valency of p is the number of edges containing p (equivalently, the number adjacent to p).

| T(p) |

of vertices

If every vertex has the same valency, the graph is called

regular, and the common valency is the valency of the graph.

(Thus, a

graph is a 0-design with k = 2; a regular graph is a 1-design.

Only the

complete graph is a 2t-design, sometimes called the pair design. ) As in design theory, we can define the incidence matrix M(T) of a graph r to be the matrix of the incidence relation (with rows indexed by edges and columns by vertices, with (e, p) entry 1 if pee, 0 other¬ wise).

A more useful matrix is the adjacency matrix A(T), the matrix of

the adjacency relation (with rows and columns indexed by vertices, with (p , p ) entry 1 if

{p , p^ }

is an edge, 0 otherwise).

Note that

M(r)rrM(r) is the sum of the symmetric A(T) and a diagonal matrix whose (p, p) entry is the valency of p; so M(T) M(T) = A(T) + al if T is regular with valency a.

Note also that A(T) = J - I - A(T), where

(as always) J denotes a matrix with every entry 1.

T is regular if and

only if A(T)J = aj for some constant a (which is then the valency of T). A strongly regular graph is a graph which is regular, but not com¬ plete or null, and has the property that the number of vertices adjacent to p^

and p^

adjacent.

(p^ ^p^) depends only on whether or not p^

and p^

are

Its parameters are (n, a, c, d), where n is the number of

vertices and a the valency, and the number of vertices adjacent to p^ and p^

12

is c or d according as p^

and p^

are adjacent or non-adjacent.

(Be warned: this notation is not standard.

We use it because a standard

notation does not exist; of the two main contenders, one is impossible here because it uses the symbols k and A, and the other is a special case of notation for association schemes and so is too cumbersome.

The

notation used here is only temporary, to be abandoned for the design theory notation whenever a design appears. ) If T is a strongly regular graph and p^

and p^

vertices, then a - c - 1 vertices are adjacent to p^ so (n - a - 1) - (a - c - 1) are adjacent to neither p remarks hold if p^

and p?

are non-adjacent.

are adjacent

but not to p^, and nor

Similar

So the complement T

is also strongly regular.

13

Only a few graphs are small enough to be drawn.

In drawings such

as Fig. 2. 1, a small circle represents a vertex, and an arc an edge, but two arcs may 'cross' without requiring a vertex at the crossing-point. Four strongly regular graphs are displayed in Fig. 2. 1. Larger graphs must be described verbally. graph T(m)

Thus, the triangular

(m s 4) has as vertices the 2-element subsets of a set of

cardinality m; two distinct vertices are adjacent if and only if they are not disjoint.

T(m) is strongly regular with parameters n = |m(m - 1),

a = 2(m - 2), c = m - 2, d = 4.

The fourth graph in Fig. 2. 1, which is

called the Petersen graph, is the complement of T(5).

The lattice graph

L2(m) (m > 2) has as vertex set S x S, where S is a set of cardinality m; two distinct vertices are adjacent if and only if they have a common coordinate.

2

L (m) is strongly regular, with parameters n = m ,

a = 2(m - 1), c = m - 2, d = 2. are L (2) and L2(3).

The first and third graphs in Fig. 2. 1

The disjoint union of k complete graphs on m

vertices each (k, m > 1) is also a strongly regular graph T(k, m) with parameters n = mk, .a = m - 1, c = m - 2, d = 0; indeed, any strongly regular graph with d = 0 is of this form. graph.

T(k, 2) is called a ladder

The complement of T(k, m) is called a complete k-partite graph.

The first graph in Fig. 2. 1 is the complete bipartite graph T(2, 2). q is a prime power with q = 1

If

(mod 4), the Paley graph P(q) has as

vertex set the elements of GF(q), with two vertices adjacent if and only if their difference is a nonzero square.

It is strongly regular, with

parameters n = q, a = ^(q - 1), c = i(q - 5), d = i(q - 1), and is iso¬ morphic with its complement.

The second and third graphs of Fig. 2. 1

are P(5) and P(9). If G is a permutation group on a set V, then G has a natural component-wise action on V x v.

We say G has rank 3, if it is tran¬

sitive on V and has just three orbits on V x V, the diagonal Up, P) Ip e V }

and two others O, O'.

If the rank 3 group G has even

order, it contains an involution interchanging two points, say p and q; thus the orbit containing (p, q) is symmetric, and the other orbit is also. Form a graph T with vertex set V, whose edges are the unordered pairs corresponding to ordered pairs in O. T.

G is a group of automorphisms of

Since G is transitive on vertices, T is regular; since G is tran-

14

sitive on adjacent pairs and on non-adjacent pairs of vertices, T is strongly regular.

The many known families of rank 3 groups of even

order yield large numbers of strongly regular graphs, including all those already mentioned; such graphs are sometimes called rank 3 graphs. Let A be the adjacency matrix of the strongly regular graph r with parameters

(n, a, c, d).

of vertices adjacent to p^

The (p^ p ) entry of A2

is the number

and p^; this number is a, c, or d according

as pi

and p^

are equal, adjacent, or non-adjacent.

(2. 1)

A2 = al + cA + d(J - I - A).

So

Also, (2. 2)

AJ = JA = aj. Thus the matrices I, J, A span a real algebra of dimension 3

which is commutative and consists of symmetric matrices.

Indeed, a

strongly regular graph could be defined as a graph whose adjacency matrix satisfies (2. 1) and (2. 2).

There is an orthogonal matrix which

simultaneouslv diagonalise£ I, J and A.

A has the eigenvalue a with

multiplicity 1, corresponding to the eigenvalue n of J; thus (2. 3)

a(a - c- 1) = (n - a - l)d,

an equation which can also be verified by selecting a vertex p2 counting the number of edges adjacent to p . values p^, p^

{p2, p? ]

with p?

and

adjacent and p,

non-

Any other eigenvalue of J is zero; so the other eigen¬ of A satisfy the equation

p2 = (a - d) + (c - d)p , whence Px, P2 = £[c - d ± V((c - d)2 + 4(a - d))] . If p

and p^ n = f

have multiplicities ^ + f

and f2> then

+1,

0 = Tr(A) = a + f: px + f2P2 .

15

These equations determine f^

and f

(since c = d = a is not possible).

Indeed we find without difficulty that

(2.4)

i[„. u v

is strongly regular. Indeed, easy counting arguments give us the parameters: b

v(v - 1) k(k - l)

k(r - 1) c — (r - 2) + (k - 1)' d

vk2kI+1 + v, and not the null graph, since r > 1.

Notice that the class of strongly regular graphs obtained

in this way includes the triangular graphs: they are the line graphs of the 'pair designs' with k = 2.

It will come as no surprise to design theorists

to learn that the rationality condition gives no extra information in this

19

case.

We shall soon see the real reason for this. (3. 1) was generalised by Bose [11] to 'partial geometries', a

class of 1-designs possessing a point graph and a line graph, both strongly regular.

However we pursue a different generalisation, due to Goethals

and Seidel [30],

In an arbitrary quasi-symmetric design with intersection

numbers x and y (x < y), define the block graph to be the graph whose vertices are the blocks, with two vertices adjacent if and only if their intersection has cardinality y. (3. 2)

Theorem.

The block graph of a quasi-symmetric design is

strongly regular. Proof.

We prove this by matrix methods resembling those used

for Fisher's inequality (1. 5).

Dr. C. Norman observed that a simple

counting argument will suffice to prove (3. 2); but the present proof has the advantage of giving us the multiplicities.

Let N be the incidence

matrix of the design, and A the adjacency matrix of its block graph. We have NTN = (r - A)I + AJ, NNT = kl + yA + x(J - I - A). T From the first equation, j is an eigenvector of N N with eigenvalue T i ± r - A + Av = rk; and N N|j has only the one eigenvalue r - A. Since T jN = kj (with a different j!) it follows that j is an eigenvector of NN

with eigenvalue rk, and that NN

|j

has eigenvalues r - A (with

multiplicity v - 1) and 0 (with multiplicity b - v). Since x * y, A is a T linear combination of I, J, and NN , and so j is an eigenvector of A. and A | j

has just two eigenvalues.

Thus the graph is strongly regular.

Its parameters can be computed from the eigenvalues and their multi¬ plicities, as described in the last section.

Notice that the multiplicities

have automatically turned out to be positive integers; this is why the rationality conditions are empty.

However, the expressions for the

parameters and eigenvalues of the block graph do give non-trivial divisi¬ bility conditions on the design parameters,

20

/

Goethals and Seidel went on to examine some special strongly regular graphs, to see whether they could be the block graphs of quasisymmetric designs. r.

Suppose r is the block graph of 2), and we know

Clearly we have the parameters v and b of 2).

meters of 2)

The other para¬

are not determined, but from expressions for the eigen¬

values of T we have relations connecting them, and these are sufficient in some cases. (3. 3)

In particular,

Proposition.

(i) If

r is a ladder graph then 2) is obtained from

a symmetric design by counting each block twice. (ii)

K T is the complement of ladder graph then 2) is a

Hadamard 3-design. (iii)

T is never a lattice graph or the complement of one.

We shall prove (ii) later, using only the knowledge of b and v. A result closely related to (3. 2) is (3. 4)

Proposition.

In a quasi-symmetric design (without repeated

blocks), b < iv(v - 1). Proof.

Let M(eq, e^, e^) be a matrix with rows indexed by

pairs of points and columns by blocks, with ( {pi, p if

I ipi> P2 } r B| = i, for i = 0, 1, 2. MT(0, 0, l)M(eo,

}, B) entry e.

We find that

, 0) = (y(k-y)ei + (k~y) eQ) A + (x(k-x)e^ + (k2X) eQ) (J - I - A).

Suppose b > |v(v - 1).

Then this matrix product is singular, for all

choices of

An eigenvector with zero eigenvalue is also an

and

eigenvector for J and A, and is clearly not j.

So if cv is the corres¬

ponding eigenvalue of A, we have with (eQ, e^) = (0, 1) and (1, 0) respectively, y(k - y)a + x(k - x)(-l - a) = 0 (k~2y)a + (k'X) (-1 - a) = 0.

21

a? = 0 = — 1 — cn is impossible, so the determinant of these equations must be zero.

But this determinant is j (k-x)(k-y)(k- l)(x-y); so

k = x or k = y or k=l or x = y.

The first two imply that the design

has repeated blocks, while the last two are impossible. J Remark. used here.

Note how little actual information from (3. 2) has been

We shall see a much more general result in Chapter 15.

Stronger results hold in special cases. (3. 5)

Proposition.

Thus

In a quasi-symmetric design 23 with intersection

numbers x and y, (i)

x = 0 implies b < v(v - l)/k;

(ii)

x = 0, y = 1 implies b = v(v - l)/k(k - 1).

Proof.

We have already seen the second part.

For the first,

suppose x = 0 and consider the incidence structure whose 'points' are the blocks of 23 containing p and whose ’blocks’ are the points of 23 different from p, with incidence the dual of that in 2). lie in y - 1

’blocks’, and any ’block’ has X ’points’, so this structure

is a 2-design. the result.

Any two 'points’

Applying Fisher's inequality to it, v - 1 > r, which gives

Furthermore, if equality holds, then the incidence structure

is a symmetric 2-design, whence its dual is also a 2-design and 2) is a 3-design.

We shall meet such designs again in the next chapter. J

Petrenjuk [52] showed that b > |v(v - 1) in a 4-design.

From

work of Delsarte, we get a characterisation of designs achieving this bound. (3. 6)

(See Chapter 15. ) Proposition.

For a 2-design 23 with 4 < k < v - 4, any two

of the following imply the third: (i)

23

is quasi-symmetric;

(ii)

23 is a 4-design;

(iii)

b = |v(v - 1).

We do not even need to assume 23 is a 2-design!

The pair

designs satisfy (i) and (iii) of (3. 6) but are clearly not 4-designs.

In

fact, up to complementation there is a unique known design satisfying all

22

the conditions of (3. 6), namely the unique 4 - (23, 7, 1) design.

N. Ito

is reported to have shown that no others exist. This 4-design provides us with several more examples of quasisymmetric designs.

If 2)

is a t-design, D

(resp. Dp) denotes the P incidence structure whose points are the points of 2) different from p, and whose blocks are the blocks of 2) incident (resp. non-incident) with p; Dp and DP are both (t-1)-designs.

If D denotes the 4 - (23, 7, 1)

design and p and q are distinct points of D, then D , Dp, Dq and pq P P D^ are all quasi-symmetric. (Dp^ *s fact symmetric, being the projective plane PG(2, 4).) D^ achieves the bound of (3. 5)(i).

The

blocks of D^ (resp. DpC*) form an orbit of PSL(3, 4) on hyperovals (resp. Baer subplanes) in the plane of order 4. The designs give us strongly regular graphs on 253, 77, 176, 56 and 120 vertices.

The first of these, together with the triangular graph

T(23), shows that the multiplicities of the eigenvalues of a strongly regular graph do not determine the parameters of the graph, even when it is known to be the block graph of a quasi-symmetric design. By contrast, we give £n example of a better-behaved situation. (3. 7)

Theorem.

A quasi-symmetric design with b = 2v - 2 is either

a Hadamard 3-design or the unique 2 - (6, 3, 2) design. Proof.

The eigenvalues of the block graph have multiplicities

1, v - 1, b - v = v - 2.

So (2. 5) applies.

Note also that v = 2k,

b = 4k - 2, r — 2k - 1.

Given a block, count the number of choices of a

point in that block and another block containing that point: ay + (4k - 3 - a)x = 2k(k - 1). If the graph or its complement is a ladder graph, we can assume a = n - 2, and so 4(k - l)y + x = 2k(k - 1). Thus 2(k - 1) divides x; since x < k we have x = 0, y = |k.

By the

remarks after (2. 5), D is a 3-design, and so a Hadamard 3-design.

23

Otherwise, we have v-l = 2s

2

+ 2s + 1, k = s

2

+ s + 1, a = s(2s + 1),

and so s(2s + l)y + (s + l)(2s + l)x = 2s(s + l)(s2 + s + 1). Thus 2s + 1 divides 2s(s + l)(s

2

+ s + 1), which implies s = 1.

It

is easy to show that there is a unique design with these parameters. (3. 3) (ii) is a consequence of this result. \l A result of Dembowski (T24] p. 5) asserts that a symmetric 2design cannot be a 3-design.

Suppose we consider the class of symmetric

designs in which the number of blocks containing three points takes just two distinct values, say x* and y*.

This class contains all symmetric

designs with A < 2, all the projective geometries, large numbers of designs with v = 2^m, and the complement of each of its members; so a complete determination seems difficult. made.

But some progress can be

Thus, for any point p, the blocks containing p and points differ¬

ent from p form a quasi-symmetric design 23* with parameters v* = k, b* = v - 1, etc. (3. 8)

Proposition.

From this we can show

A Hadamard 2-design in which any three points lie

in x* or y* blocks is either a projective geometry over GF(2) or the unique 2 - (11, 5, 2) design. Proof.

If 23 is the complement of the given design, then 25*

satisfies the hypotheses of 2. 7; if 23* is a Hadamard 3-design for any point p it is easy to show that the original design is a projective geo¬ metry.

y Note also that if x* = 0 then 23* achieves the bound of (3. 5)(i).

Using result (1. 12), we can obtain strong information about the parameters of such a design. It can also be shown that (3. 9)

Proposition.

If 23

is a symmetric design in which any 3 points

lie in 0 or y* blocks, then the same is true of the dual of 23. It is not known whether this remains true when 0 is replaced by an arbitrary x*.

24

4 • Strongly regular graphs with no triangles

In a graph, a path is a sequence of vertices in which consecutive vertices are adjacent, and a circuit is a path with initial and terminal point equal.

A graph is connected if any two vertices lie in a path.

The

function d defined by d(p, q) = length of the shortest path containing p and q, is a metric in a connected graph; the diameter of the graph is the largest value assumed by d.

The girth of a graph is the length of the

shortest circuit which contains no repeated edges, provided such a circuit exists. For strongly regular graphs, the connectedness, diameter and girth are simply determined by the parameters.

T is connected with

diameter 2 if d > 0, and is disconnected if d = 0.

T has a girth pro¬

vided a > 1; the girth is 3 rf c > 0, 4 if c = 0 and d > 1, and 5 if c = 0, d = 1. It is easy to see that a graph with diameter 2 and maximal valency a has at most a

2

+ 1 vertices; and a graph with girth 5 and minimal

valency a has at least a

2

+ 1 vertices.

Equality holds in either case

if and only if the graph is strongly regular with c = 0, d = 1. graph is called a Moore graph with diameter 2.

Such a

It is worth noting that

analogous bounds exist for larger values of diameter and girth, but recently Bannai and Ito [7] and Damerell [201 have shown that they are attained only by graphs consisting of a single circuit.

In the diameter 2

case, however, non-trivial examples do exist. The condition c = 0, d = 1 is so strong that the rationality con¬ dition can be expected to give useful information. ^[a

±

^1 are integers.

Indeed it shows that

Type I occurs precisely if a= 2; here

the pentagon is the unique graph.

In type II we have a = i(u2 + 3) for

some integer u which divides —(u 1 6

+ 3)(u

- 5); so u divides 15.

u = 1 is not possible, so u = 3, 5 or 15, a = 3, 7 or 57, n = 10, 50 or 3250.

25

This was first proved by Hoffman and Singleton [38], who went on to analyse the first two cases.

If n = 10, a = 3, it is easy to see that

the Petersen graph is the unique example.

Hoffman and Singleton were

able to show the existence and uniqueness of a graph with n = 50, a = 7. The final case is too large for any complete analysis so far. Since this case is fairly well settled, we shall exclude it and con¬ sider strongly regular graphs of girth 4.

These include the complete

bipartite graphs (with c = 0, d = a), which are uninteresting, so we ex¬ clude them as well and assume c = 0, 1 < d < a. (4. 1)

Theorem.

1 < d < a.

Let T be a strongly regular graph with c = 0,

Let p be any vertex, and let 25(T, p) denote the incidence

structure with point set T(p) and block set T(p), with a point and block incident if and only if they are adjacent in T.

Then 25(T, p) is a 2-

design with v = a, k=d, A = k - 1, r = v - 1, b = v(v - l)/k, possibly having repeated blocks. Proof.

Clearly any block is incident with d points.

points of 25(T, p).

Take two

They are both adjacent to p, and so are not adjacent

to each other, and d - 1 more vertices are adjacent to both, all of which must lie in T(p).

The other parameters follow. J

For the remainder of this chapter, we will allow designs to have repeated blocks, and will use the design parameters v and k in place of a and d.

Now we can ask: which 2-designs with A = k- 1

are

expressible as 25(T, p) for some strongly regular graph T of girth 4? The rationality conditions give us some information, though it is not as strong as in the Moore graph case.

They show that

V(k2 - 4k + 4v) are integers.

Type I cannot occur (d * c + 1), so k2 - 4k + 4v is a

square, say k2 - 4k + 4v = (k + 2s)" where s is an integer (since k2 - 4k + 4v = k2 (mod 2)), s > 0.

26

Then

(4. 2)

Proposition,

v = (s + l)k + s2, and

JLr ((s+l)k+s2)((s+2)k+s2-l) ((s + l)k+s2)((s+2)k+s2-3) , 21 k ±-k+2i-J are integers. In particular, the two terms inside the square brackets are both integers, whence k divides s2(s"-l) and k+2s divides s(s + l)(s + 2)(s+3). The last condition implies that k+ 2s divides k(k- 2)(k- 4)(k- 6), so for any value of k different from 2, 4 or 6, there are only finitely many possible values of v.

This was observed by Biggs [91.

For k = 2,

4 or 6, the rationality condition allows infinitely many values of v.

For

example, if k = 2, then v = (s + l)2 + 1, and s + 1 is not divisible by 4. Having chosen a 2-design with A = k - 1 as a potential 23 (T, p), we know all the edges of T except those joining distinct blocks.

(Thus

the problem resembles that of extending a given design.) Any block must be adjacent to v - k others, and adjacent blocks must be disjoint (since the graph contains no triangles); furthermore, if two blocks are nonadjacent and have x common points, then k - x blocks must be adjacent to both.

Sometimes it happens that exactly v - k blocks are disjoint

from any given block.

Then there is no choice in the construction; two

blocks are adjacent if and only if they are disjoint.

Two examples of

this, where the resulting graph is strongly regular, are the pair design on five points, and the unique

3 - (22, 6, 1) design.

The graphs obtained

are the complement of the Clebsch graph (n = 16) and the Higman-Sims graph (n = 100) respectively.

In fact, only two further examples of

strongly regular graphs of girth 4 (other than complete bipartite graphs) are known.

These are the graphs on 56 and 77 vertices constructed in

the last chapter.

The corresponding designs are the pair design on 10

points and the 4-dimensional affine symplectic geometry over GF(2). (See Chapter 1.) The rule for adjacency of blocks in the first case can be expressed in terms of the cross ratio, assuming that the ten points are identified with a projective line over GF(9); in the second case, blocks are adjacent if they are disjoint but not parallel. If p and q are adjacent vertices in the Higman-Sims graph T,

27

then r|T(p) and r|(T(p) n T(q)) are the graphs on 77 and 56 vertices (Similarly, if p and q are adjacent vertices in the

just described.

complement of the Clebsch graph, then r| T(p) is the Petersen graph, and r| (T(p) n T(q)) is a ladder graph.) In addition, the Petersen graph can be partitioned into two pentagons, and the Higman-Sims graph into two Hoffman-Singleton graphs.

(The last fact was proved by Sims [63].)

We shall see later the reason for some of these facts; others, however, remain slightly mysterious. In the situation of a strongly regular graph with girth 4, the observation that adjacent blocks are disjoint gives the following result. (4. 3)

Theorem.

Let T be a strongly regular graph with a = v, c = 0,

d = k, and 1 < k < v.

Let D(r, p) be the 2-design of (4. 1),

Then

v > i[3k + 1 + (k - 1) V(4k + 1)]. If v = (s + l)k + s2 Proof.

(as in (4. 2)) then k < s(s + 1).

By (1. 7), the number of blocks disjoint from a given block

• 2.

Then the

following are equivalent: (i)

2D(r, p) is a 3-design;

(ii)

2D(T, p) is quasisymmetric;

(iii)

r|T(p) is strongly regular;

(iv)

v - i[3k + 1 + (k - 1) V(4k + 1)].

Any one of these implies that v = s(s2 + 3s + 1), k = s(s + 1), and that r|T(p) is the complement of the block graph of 2D(T, p). Remark.

Before proving this, we note that the condition k ^ 2

is essential, as the graph on 56 vertices shows. Proof.

By (4. 4), (i) and (ii) are equivalent (since b = v(v-l)A).

If either holds, then (1. 12) gives us rather limited possibilities for the parameters of 2D(T, p):

29

(i)

v = 4y, k = 2y;

(ii)

v = y(y2 + 3y + 1), k = y(y + 1);

(iii)

v = 112, k = 12, y = 2;

(iv)

v = 496, k = 40, y = 4.

Of these, the first is excluded by (4. 3), and the third and fourth fail the rationality condition (4. 2).

We are left with v = y(y

+ 3y + 1),

k = y(y + 1), for which (iv) is easily verified. If (iv) holds then we have equality in (4. 3), and so two blocks are adjacent if and only if they are disjoint, while non-disjoint pairs of blocks have equally many common points.

So (ii) and (iii) hold

Finally, suppose (iii) holds.

Then any two non-adjacent blocks are

both adjacent to t further blocks, and so are adjacent to (that is, contain) k - t points, for some integer t; adjacent blocks are disjoint.

Thus

2D(T, p) is quasi-symmetric and r|r(p) is the complement of its block graph, sj Remark.

The proof of (1. 12) proceeds as follows.

Let 2D be

an extension of a symmetric design, and B a block of 2D.

First it is

shown that the points outside B and blocks disjoint from B form a 2- design, which is degenerate only in the case where 2D 3- design.

is a Hadamard

Thus, if 2D is non-Hadamard, Fisher's inequality (1. 5) shows

that at least v - k blocks are disjoint from B, and so the inequality of (4. 3) holds.

From this, the parameter sets given in (1. 12) are found

using only the divisibility conditions.

Here, we already have (4. 3), and

so the proof of (1. 12) can be considerably shortened. If r satisfies the hypotheses of (4.5), then 2D(T, p) is a quasisymmetric 3-design, and so 2D(T, p)^ is a quasisymmetric 2-design for any point q.

Thus, if p and q are adjacent vertices, rjr(p) and

r I (T(p) n T(q)) are strongly regular graphs containing no triangles.

The

parameters of the corresponding designs are 2 - (s (s + 2), s , s - 1) and 2 - (s(s

+ s - 1), s(s - 1), s

are non-adjacent, (T(p)

- s - 1).

Note also that if p and r

n T(r), T(p) n T(r)) is a symmetric 2-design with

parameters (s (s + 2), s(s + 1), s), with incidence = adjacency in T. Let us exhibit one corollary of (4. 5), concerning extendability of designs.

30

(1. 12) shows that the parameters of an extendable symmetric

design are of one of the forms (i)

(4A + 3, 2A + 1, A)

(Hadamard 2-design);

(ii)

((A + 2)(A2 + 4A + 2), A2 + 3A + 1, A);

(hi)

(111, 11, 1);

(iv)

(495, 39, 3).

It is known that any Hadamard design is uniquely extendable. true for the other parameter sets.

This is not

Let us consider case (ii).

With

A = 1, there is a unique design, namely PG(2, 4), and it possesses three different (though isomorphic) extensions.

With A = 2, Hall, Lane,

and Wales [34] have constructed a symmetric 2 - (56, 11, 2) design, but Baumert and Hall have shown it is not extendable (M. Hall Jr., personal communication). (4. 6)

Theorem.

However, a weaker result is true:

If a symmetric 2 - ((A+2)(A2 + 4A+2), A2 + 3A+1, A)

design is extendable, then so is its dual. Proof.

Let 33 be a 3-design and q a point of 33 for which 33^

is isomorphic to the given symmetric design.

In 33, exactly v - k

blocks are disjoint from anyjjlock, and there is a strongly regular graph T of girth 4 with a vertex p such that 33(T, p) = 33.

Since (iv) of (4. 5)

is independent of the vertex p, it follows that 33(T, q) is also a 3-design; and it is easy to see that 33(T, q)^ is the dual of the given symmetric design, y Noda [51] proved some of the results in this section from a differ¬ ent approach.

He showed that if a strongly regular graph with c = 0

and 1 < d < a has the property that exactly s vertices are adjacent to each of three pairwise non-adjacent vertices, then a = s(s d = s(s + 1).

+ 3s + 1),

From (4. 5) we can deduce the same conclusion from the

weaker hypothesis that exactly 0 or s vertices are adjacent to each of any three vertices, provided d > 2.

(The graph on 56 vertices is a

counterexample with d = 2.) Indeed, we need only require the condition for triples containing a particular vertex.

Noda further showed that if,

in addition, the automorphism group of T is transitive on triples of pairwise non-adjacent vertices, and if s is a prime power, then s=l or 2 (and the graph is known).

The condition on s was weakened in

31

Cameron's thesis, but has not been removed. If T is an arbitrary strongly regular graph, then

£>(r,

p) is a

1-design, (provided only that d > 0), but it is not in general a 2-design. (The parameters are v = a, b = n - a - 1, k = d, r = a - c - 1. ) Sup¬ pose c> 0 and

2)(r,

p) is a 2-design.

Then two points of T(p) are

adjacent to A points of T(p), whether or not they are adjacent; so they are adjacent to c - A - 1 or d-A-1 points of T(p), according as they are adjacent to each other or not.

Thus

rj T(p) is strongly regular

Applying 'a(a - c-l) = (n-a - l)d' to it, we have cA = (a-c-l)(d-A-l), whence (a - c - 1)(d - 1)

Thus (4. 7)

Theorem.

Let r be strongly regular, with parameters

(n, a, c, d), where c> 0 and d > 0.

Then T>(T, p) is a 2-design if

and only if T [ T(p) is. strongly regular with parameters (a, c, c - d +

c(d - 1) a - 1

c(d - 1) a - 1 )

.

An example of such a graph T was found by Paulus and Seidel (J. J. Seidel, personal communication).

T has parameters (26, 10, 3, 4);

T T(p) is

a Petersen graph, with parameters (10, 3, 0, 1); and 25(r, p) is a 2 - (10, 4, 2) design.

Indeed, Paulus found six non-isomorphic graphs

with these properties and parameters.

32

5 ■ Polarities of designs

Let T be a strongly regular graph in which n = v, a = k, c = d = A.

Then any two vertices are adjacent to exactly A others.

Such a graph is called a (v, k, A)-graph (among other things).

Clearly

a symmetric 2 - (v, k, A) design can be constructed from it as follows: the points and blocks are both indexed by the vertices of T, and a point and block are incident if and only if their labels are adjacent.

This

design has a further remarkable property: the correspondence between points and blocks indexed by the same vertices is a polarity with no absolute points.

Not surprisingly, the existence of such a polarity is a

very strong restriction on the design.

The rationality condition shows

that

are integers.

2

2

Putting k - A = u , we see that u divides k = A + u ,

and so u divides A.

Thus there exist only finitely many (v, k, A)-graphs

with a given value of A.

In the extremal case with A = u we have

k = A(A +1), v = A (A + 2).

Ahrens and Szekeres [1] showed the existence

of a graph with these parameters for each prime power value of A; indeed, their graphs are line graphs of generalised quadrangles.

The

smallest examples are the (degenerate) triangle (A = 1), L (4) and another graph with the same parameters characterised by Shrikhande [61] (A = 2), and the graph whose vertices are the 45 intersection points of the 27 lines on a general cubic surface, adjacent if they are collinear (A = 3).

The fact that the triangle is the only (v, k, 1) graph has appeared

in different guises.

It is the Friendship Theorem of Erdos, Renyi and

Sos [25], for which several elementary proofs (avoiding use of the ration¬ ality conditions) have recently been given.

It is also the fact, due to Baer,

that a polarity of a finite projective plane has absolute points.

([24]

p. 152.) 33

A graph in which any two vertices are adjacent to A vertices is either a (v, k, A) graph or one of the 'windmills' of Fig. 5. 1 with A = 1.

Fig. 5. 1 If a symmetric 2 - (v, k, A) design has a polarity a with no absolute points, we can recover the (v, k, A) graph as follows: the vertices are the points, and p and q are adjacent if q e pa.

(This

relation is symmetric since a is an involution, and irreflexive since a has no absolute points; the other conditions follow from the design properties.) So the two concepts are equivalent. Now, let T be a strongly regular graph in which n = v, a = k - 1, c = A - 2, d = A.

Construct an incidence structure as follows: points

and blocks are indexed by vertices, and a point and block are incident if and only if their labels are equal or adjacent.

This incidence structure

is a symmetric 2 - (v, k, A) design, and the correspondence between points and blocks with the same label is again a polarity, but this time every point is absolute.

(Such a polarity is called null. ) Again the graph

can be recovered from the design 2) and the polarity a.

(Vertices are

points of 2); p and q are adjacent if q e pa but q * p. ) This time the rationality condition shows that

34

v - k - A)

£[v - 1 2 * *

7 (k

are integers.

]

This does not give a bound for v in terms of A.

The case A = 1 cannot occur here, since d = A - 2 is non¬ negative.

This fact, clearly not peculiar to the finite case, is also a

consequence of the fact that an absolute line in a projective plane carries a unique absolute point.

A null polarity of a design induces a polarity of

the complementary design with no point absolute; so the two cases are really the same.

However, it is often convenient to separate them.

Null

polarities have received less attention than polarities with no absolute points.

An interesting case is that with A = 2.

Here d = A - 2 = 0, and

the graphs are among those discussed in the preceding chapter, where we saw that k = u

+2 for some integer u not divisible by 4.

with u = 2 and u = 3 were constructed there. determined by their parameters.

Examples

These graphs are uniquely

(This is trivial for u = 2, and is a

result of Gewirtz [26] for u = 3. ) Thus, 2 - (16, 6, 2) and 2 - (56, 11, 2) designs with null polarities exist and are unique.

The first of these is the

same one that also has polarities with no absolute points!

The next result

is easy to verify. (5. 1)

Theorem.

Let 23 be a symmetric 2-design with A = 2, and a

a null polarity of D. (i)

For any block B and any two points, p, q e B - {B*7],

p, q and B° ’generate' the trivial 2 - (4, 3, 2) design. (ii)

a is uniquely determined by the image of a point or block.

(iii)

Two distinct null polarities commute, and their product is a

fixed-point-free involution. (iv) g^

If o\ ,

, cy

are three distinct null polarities, then

is a polarity with no absolute points. It follows from a previous remark that the only non-trivial 2-

design with A = 2 admitting three distinct null polarities is the 2 - (16, 6, 2) design (which has 6 such polarities). Next we give some further examples of graphs with the above properties.

35

Suppose H is a Hadamard matrix.

We may interpret H as an

incidence matrix (with, say, -1 for incidence and +1 for non-incidence). If H is symmetric and has a constant diagonal, we can change signs to make the first row and column consist of -l's without affecting these properties; deleting them we obtain the incidence matrix of a Hadamard 2-design with parameters 2 - (4u polarity.

- 1, 2u

- 1, u

- 1) with a null

If in addition the original H has constant row sums, then it

or its negative is the incidence matrix of a 2-design with parameters 2 - (4u , 2u

± u, u

± u) with a null polarity; the sign is + or -

according as the signs of the diagonal and the row sums agree or not. Such Hadamard matrices are quite common.

Thus, one exists realising

each possibility for the signs if a Hadamard matrix of order 2u exists. They exist for other orders, such as 36, as well.

(See [29] and [16].)

Other examples of null polarities are provided by the symplectic polarities of projective (2m - l)-space over GF(q), m > 2.

Also, if a

is an orthogonal polarity of projective 2m-space over GF(q), with m > 2 and q is odd, then the absolute points and hyperplanes form a symmetric design, and a induces a null polarity of this design.

The design has the

same parameters as PG(2m - 1, q), but is not isomorphic with PG(2m - 1, q). Further examples when q = 3 arise as follows. quadratic form associated with a.

Let Q be the

Since Q(x) = Q(-x), Q induces a

function from projective points to GF(3); denote it by Q as well. P.

If

is the set of points on which Q takes the value i (i = ±1), then

(P., P^) is a symmetric design, and a induces a polarity of it with no absolute points.

The parameters of this design are

0 ,i „m._m , .\ i ,m-l,,m .\ i Qm-l,„m-l 2 - (l 3 (3 + l), i3 (3 - i), ¥ 3 (3

i)), i = ±1,

provided the sign of Q is chosen correctly. Just as there exist 'free' symmetric designs with arbitrary finite values of A (and indeed much more general incidence structures), so there exist 'free' graphs in which the number of vertices adjacent to p and q is c or d according as p and q are incident or not, where c and d are prescribed non-negative integers with d > 1.

Indeed, the

construction is even simpler; vertices and edges are added together at

36

each step.

In particular, there are free symmetric designs with polari¬

ties with no absolute points and arbitrary A, and free symmetric designs with null polarities and arbitrary A > 1. polarity with A = 2

Taking the case of a null

(that is, c = 0, d = 2), we have a design in which

infinitely many instances of an 'incidence theorem’ hold (see (5. l)(i)).

✓

37

6 • Extension of graphs

In this section we shall be considering graphs from a slightly different point of view, as incidence structures in their own right (as remarked in Chapter 2).

In particular, a regular graph is a 1-design

with k = 2, and conversely (provided we forbid repeated blocks).

An

interesting question is: when do such 1-designs have extensions?

Since

2- designs with k = 3 are so common, this problem is too general, and we shall usually impose extra conditions on the extension.

Extensions of

designs were originally used by Witt, Hughes and Dembowski for studying transitive extensions of permutation groups, and it might be expected that extensions of regular graphs would be useful in the study of doubly transitive groups.

This is indeed the case.

If we are extending a design by adding an extra point p, we know all the blocks of the extension containing p - these are obtained simply by adjoining p to all the old blocks - but it is in the blocks not containing p that possible ambiguities arise.

Let us then suppose, as a first possi¬

bility, that we have a set A of triples, called blocks, in which the blocks not containing a point p are uniquely determined in a natural way by those containing p; in particular let us suppose that, when we know which of {pqr),

{pqs }

is a block.

and

{prs }

are blocks, we can decide whether

{qrs }

It is easy to see that the rule must be that, given a 4-set

{pqrs ), the number of its 3-subsets which are blocks belongs to a sub¬ set S of

{0, 1, 2, 3, 4 }

with the property that i, j e S, i*j=> |i-j|>2.

Further analysis shows that S is a subset of one of or

{0, 2, 4 },

{1, 4 },

{0, 3 ), and that in the second or third case either A or its comple¬

ment is the set of all triples containing a particular point, an uninteresting situation. A set A of triples, with the property that an even number of the 3- subsets of any 4-set belong to A, was called a two-graph by G. Higman,

38

who first studied such sets.

Two-graphs turn out to be fascinating ob¬

jects, closely connected with graph theory, finite geometry, real geom¬ etry (sets of equiangular lines in Euclidean space), etc.

In addition, a

large number of the known finite doubly transitive groups act on twographs. odd, V

(These include PSL(2, q) for q = l(mod 4), PSU(3, q) for q

Go(q), Sp(2n, 2) in both doubly transitive representations,

nSp( n, 2), HS, and . 3. ) Two-graphs were studied extensively by

2

2

D. Taylor and J. J. Seidel, and many others contributed to the theory. Seidel [59] presented a survey at a recent conference in Rome, so only a brief sketch will be given here. Clearly a two-graph can be regarded as a map from the triples on a set into

with vanishing coboundary, that is, a 2-cocycle.

Since

all relevant cohomology is trivial, it is thus the coboundary of a 1-cochain, the latter being a function from pairs into Z^, that is, a graph; the triples of the two-graph are those carrying an odd number of edges of the graph.

Two 1-cochains have the same coboundary if and only if they

differ by a 1-cocycle, that is the coboundary of a 0-cochain (a function from the point set to Z^); this means there is a partition of the point set as X U Y, and the two graphs agree within X and within Y and are complementary between these sets.

In Seidel's terminology, the graphs

are related by switching with respect to the partition X u Y. two-graph is equivalent to a 'switching class' of graphs.

Thus a

Seidel uses

(0, -, +) adjacency matrices for graphs (these have 0 on the diagonal, -1 for adjacency, and +1 for nonadjacency). A, A’

The adjacency matrices

of switching-equivalent graphs are related by A’ = DAD, where

D is a diagonal matrix with diagonal entries ±1.

The null two-graph

corresponds to the switching class of complete bipartite graphs, and the complete two-graph to that of disjoint unions of two complete graphs. These will be excluded in future. If A is any set of triples on P and p is any point of P, let A(p) be the graph with vertex set P - {p } {{q, r } | {p, q, r } e A }.

and edge set

It is easy to see that, if A is a two-graph,

then the switching class corresponding to A contains the disjoint union of the isolated vertex p and A(p).

(Take any graph in the class and

39

switch with respect to the neighbours of p. ) Suppose A(p) is regular. A(p) determines A uniquely; when is A an extension of A(p) in the design sense, that is, when is A a 2-design? A two-graph which is also a 2-design is called a regular two-graph. (6.1)

Theorem.

The precise answer is:

A two-graph A is regular if and only if, for some

point p, A(p) is strongly regular with d = ia.

If this holds for one

point, then it holds for every point. Using this and the rationality conditions, one sees that the para¬ meters of regular two-graphs are fairly restricted (though still general enough to include all the examples with doubly transitive automorphism groups mentioned earlier).

Any strongly regular graph of Type I satis¬

fies the condition of (6. 1); if A(p) is the Paley graph P(q) then A admits the doubly transitive group PSL(2, q). Seidel also observed (6.2)

Theorem.

A two-graph is regular if and only if the (0, -, +)

adjacency matrix of some (and hence every) graph in its switching class has just two eigenvalues. Note that the equation (A - piI)(A - p^l) = 0 is invariant under switching.

This equation implies that, if the graph is regular, then it is

strongly regular.

It is not difficult to show that there are two possible

sets of parameters for a strongly regular graph in the switching class of a regular 2-graph.

Thus, for example, there is a unique (up to

complementation) regular two-graph on 28 points, with design parameters 2 - (28, 3, 10); its automorphism group is Sp(6, 2).

The possible para¬

meters for strongly regular graphs in the switching class are (28, 15, 6, 10) and (28, 9, 0, 4).

The first set is realised by the com¬

plement of the triangular graph T(8), and also by three exceptional graphs characterised by Chang [18], [19], set is impossible.

We saw in section 3 that the second

Thus both possibilities do not always occur, though

they do in many interesting cases. Briefly, the connection with equiangular lines is as follows.

Given

a set of lines through a point of Euclidean space, with any two at an angle with cosine ± a(oi ± 0), the set of triples of lines for which the product

40

of cosines is negative forms a two-graph; conversely any two-graph can be represented in this way.

The 'best' known sets of equiangular lines

(with most lines in space of a given dimension) often represent regular two-graphs; the theory of regular two-graphs also gives upper bounds for the number of lines.

(See Lemmens and Seidel [43].)

E. Shult [62] proved the following result on transitive extensions: (6. 3)

Theorem.

phism group G. elements

Let T be a graph with a vertex-transitive automor¬ Suppose that, for some vertex p of r, there exist

e Aut(r| r(p)) arid h2 eAut(r|T(p)) such that, for

q e T(p), r e T(p), {q, r} £

{q, r } e

{qh^, rh^ } e T.

Proof.

{qh^ rho } £ T and Then G has a transitive extension.

If A is the two-graph for which A(p ) = T for some

vertex pQ, then it is easy to see that the permutation which interchanges p.

and p and acts as h^

on T(p) and h^

on T(p) is an automorphism

of a. y More generally, one could define a t-graph to be a t-cocycle over *

Z^, and a regular t-graph to be a t-graph which is also a t-design. t > 2, however, examples seem to be much more limited.

For

The only ones

we know are regular 3-graphs on 8 and 12 points (the former is AG(3, 2), and the latter a double extension of the Petersen graph with automorphism group M

discovered by D. G. Higman) and a regular 5-graph on 12

points (the 5 - (12, 6, 1) design).

Indeed it is trivial that any

t - (2t + 2, t + 1, 1) design is a regular t-graph; but only two such designs are known.

Also, little machinery for studying regular t-graphs

has been developed, apart from D. G. Higman's generalisation of (6. 3) to t-graphs [37].

Again, one could generalise by considering cocycles

over other abelian groups, but here too examples are fairly scarce and general theory is lacking. A two-graph could be defined as a set A of triples such that A and its complement both satisfy the following condition: (*)

If

Ipqr],

{pqs },

Iprs } f A then

{qrs } f A,

41

It seems that if A only (and not its complement) is assumed to satisfy (*), the methods used for two-graphs fail, but the structure of A is sufficiently restricted that some progress is possible.

A useful

remark is: (6. 4)

Let T be a graph on v - 1 vertices in which the

Lemma,

largest complete subgraphs (cliques) have size k - 1, any vertex lies in A of them, and any edge lies in at least one of them.

Suppose A is a

set of triples satisfying (*), with A(p) = T for any point p.

Then there

is a 2 - (v, k, A) design D, such that A is the set of triples which are contained in some block of D. The blocks are the sets S with

|s|

= k, all of whose triples

belong to A; these are the sets S for which

|s|

clique in A(p) for some (and hence every) p e S. lie in A such blocks, and a triple it is in A.

- {p }

is a (k-1)-

Clearly two points

is contained in a block if and only if

We shall say the design represents the set of triples.

Note

that T satisfies the hypothesis of (6. 4) if it admits a vertex- and edgetransitive automorphism group. (6. 5)

Theorem.

A few applications follow.

Suppose T is a graph, and A a set of triples satis¬

fying (*) with A(p) = T for any point p. (i)

If T is a disjoint union of complete graphs on k - 1 ver¬

tices, then A is represented by a 2 - (v, k, 1) design, and conversely. (ii)

If T is a triangular graph T(k), then A is represented by

a symmetric 2 - (v, k, 2) design satisfying the following axiom: if B is a block and p, q, r e B, then the 'second blocks' through pq, qr and rp are concurrent.

(Thus p, q, r generate the 2 - (4, 3, 2) design. )

The converse is also true. (iii)

II T is a lattice graph Lo(m), then m= 3 and A is the

unique regular two-graph on 10 points. Proof.

The proof of (i) is trivial.

In (ii) note that for k > 4 the

triangular graph T(k) has two kinds of cliques, viz. { lk }}

and

{{12},

{ 13 },

{23}}.

The first extend to blocks, the second

to point sets of 2 - (4, 3, 2) designs.

42

{{12 }, { 13 }, .. . ,

(The case k = 4 can be done by

hand. ) For (iii) we have v = m2 + 1, k = m + 1, A = 2; so r = 2m, , 2m(m2 + 1) , _ . b = —^ + i—, whence m = 3, and uniqueness is easy. 7 The only known design with k > 3 satisfying the axiom of (6. 5) (ii) is a 2 - (16, 6, 2) design which we met in the last section; for this design, A is a regular two-graph! a difficult problem. design 2D^ (6. 6)

Characterising it by this axiom seems

We mention a weaker characterisation.

Call this

for the moment.

Theorem.

If 33

is a symmetric 2 - (v, k, 2) design and B a

block of 3D such that any four points of B generate 33, , then 33 = 33, . -

—-

-

1 6

16

-

This has as a consequence a result on transitive extensions, closely related to extensions of triangular graphs: (6. 7)

Theorem.

Let H be a permutation group of degree k inducing

a rank 3 automorphism group of T(k), and G a transitive extension of this rank 3 group.

Then one of the following holds:

(i)

k = 4, H = S , G = PSI (2, 7)

(ii)

k = 5, H = A , Cr = PSL(2, 11)

(iii)

k = 6, H = A

or S , G = V

.A

orV

. S .

In each case G acts on a symmetric 2 - (v, k, 2) design.

43

7- Codes

In coding theory one considers a set F of q distinct symbols which is called the alphabet. F = { 0, 1 }.

In practice q is generally 2 and

In most of the theory one takes q = p

(p prime) and

F = GF(q). Using the symbols of F one forms all n-tuples, i. e. calls these n-tuples words and n the word length.

Fn, and

If F = GF(q) we shall

denote the set of all words by (R^ and interpret this as n-dimensional vector space over GF(q). In 0 and x e (R'

.)

\

we define the sphere of

radius p with centre at x by S(x, p) := {ye (R^ | d(x, y) < p }. /

Consider a subset C of 6r

\

with the property that any two dis¬

tinct words of C have distance at least 2e + 1.

If we take any x in C

and change t coordinates, where t < e (i. e. we make t errors), then the resulting word still resembles the original word more than any of the

44

others (i. e. it has a smaller distance to x than to other words). fore, if we know C, we can correct the t errors.

There¬

The purpose of

coding theory is to study such e-error-correcting codes (e. g. coding, decoding, implementation).

In these lectures we shall be interested in

certain connections to combinatorial designs. (7. 4)

Definition.

An e-error-correcting code C

is a subset of Grn'

with the property

VxeCVyeC[x

d(x’ y) ~ 2e + ^

i. e. VxeCVyecfX

S(x’ e) n S(y’ e) = 0] •

*

A special type of code which is not too important in practice but of great interest to combinatorialists and group theorists (because of inter¬ esting automorphism groups) is a perfect code: (7. 5)

Definition.

An e-error-correcting code C in (R^ is called

perfect if

'

u S(x, e) = (R^ . xeC One of the more interesting types of codes is the linear code: (7. 6)

Definition.

A k-dimensional linear subspace C

of (R^ is called

a linear code or (n, k)-code over GF(q). The error-correcting capacity of a code is determined by the minimum distance between all pairs of distinct code words.

For an

arbitrary code consisting of K code words there are (^ ) comparisons to make to find this minimum distance.

For linear codes we have the

following advantage: (7. 7)

Theorem.

In a linear code the minimum distance is equal to the

minimal weight among all non-zero code words. Proof. w(x-y).

If x

e

C, y e C, then x - y e C and d(x, y) = d(x-y, 0) =

7

45

We shall now look at two ways of describing a linear code C. The first is given by a generator matrix G which has as its rows a set of basis vectors of the linear subspace C.

Since the property we are

most interested in is possible error-correction and this property is not changed if in all code words we interchange two symbols (e. g. the first and second letter of each code word) we shall call two codes equivalent if one can be obtained by applying a fixed permutation of symbols to the words of the other code.

With this in mind we see that for every linear

code there is an equivalent code which has a generator matrix of the form (7. 8)

G =

where I,

K

P)

is the k by k identity matrix and P is a k by n - k matrix.

We call this the standard form of G.

If one is not interested in linearity

one can extend the concept of equivalence by also allowing a permutation of the symbols of the alphabet.

A code C is called systematic if there

is a k-subset of the coordinate places such that to each possible k-tuple of entries in these positions there corresponds exactly one code word. Note that by (7. 8) an (n, k)-code is systematic. We now come to the second description of a linear code C. (7. 9)

Definition.

If C is a linear code of dimension k then

CX := {x e (R^n) | Vy fC[(x, y) = 0]}, (where (x, y) denotes the normal inner product) is called the dual code of C. The code for C

is an (n, n-k)-code.

If H is a generator matrix

then H is called a parity check matrix for C.

In general a

parity check for the code C is a vector x which is orthogonal to all code words of C and we shall call any matrix H a parity check matrix if the rows of H generate the subspace

CX.

Therefore a code C is

defined by such a parity check matrix H as follows: (7. 10) C = {x e (R(n)|xHT = 0 } .

46

Remark.

The reader is warned that often two codes, C, C*

are called duals if C* is equivalent to C"1.

This often leads to con¬

fusion but it also has certain advantages. Let us consider an important example of linear codes.

Consider

the lines through the origin in AG(m, q) and along each of these choose a vector x.

(i = 1, 2, .. . , n := (qm - l)/(q - 1)).

(m rows, n columns) with the x.

Form the matrix H

as column vectors.

Since no two

columns of H are linearly dependent we see that H is a parity check matrix of a code C in which every non-zero code word x has w(x) ^ 3 (by (7. 10)).

Clearly, this code has dimension n - m.

called the (n, n-m)-Hamming code over GF(q).

It is

The previous remark

shows that this is a single-error-correcting code. (7. 11) Theorem. Proof.

Hamming codes are perfect codes.

Let C be a (n, n-m)-Hamming code over GF(q).

n = (qm - l)/(q - 1). Since

If x e C then

Then

|s(x, 1)| = 1 + n(q - 1) = qm.

|c| = qn m and C is single-error-correcting we find that

| U S(x, 1) | = q11, i. e. C is perfect, y xeC Let C be an (n, k)-code. To every word (c , c , ..., c ) of C we add an extra letter c

(say in front) such that c + c +.. . + c =0. o J oi n This is called an overall parity check. In this way we obtain a new linear code C which is called the extended code corresponding to C.

If H is

a parity check matrix for C then

f1

1

1

...

1~]

_H = i 00

is a parity check matrix for C.

Of course if the all-one vector (which we

denote by 1) is a parity check vector for H, then the extension is trivial because c

= 0 for all words of C.

However, if C is a binary code

with an odd minimum distance d, then obviously the minimum distance of C is d + 1.

47

For our first connection to the theory of designs we consider as an example the extended (8, 4)-binary-Hamming code.

We have seen

above that *\ 111111 0 0 0 0 1 1

H :=

0 0

v_

0 110 10 10

0 1

1 1 1 0

1 1 1 1

is a parity check matrix for this code.

J

It is easily seen that the code is

equivalent to a code with generator G = (1^

- Ij.

If we make a list

of the 16 words of this code we find 0, 1 and 14 words of weight 4.

Now,

since two words of weight 4 have distance > 4, they have at most two l's in common.

It follows that no word of weight 3 is a 'subword' of g

more than one code word.

There are ( ) = 56 words of weight 3 and

each code word of weight 4 has four subwords of weight 3.

We have

therefore proved: (7. 12) Theorem.

The 14 words of weight 4 in the extended (8, 4)-binary

Hamming code form a 3-design with parameters v = 8, k = 4,

X

= 1.

We leave it as an exercise for the reader to check that this design is the extension of PG(2, 2), i. e.

AG(3, 2).

Besides as an example, the discussion above has served a second purpose namely to show that it can be useful to know how many words of some fixed weight i are contained in a code C.

A simple way to des¬

cribe such information is given in the following definition. (7. 13) Definition.

Let C be a code of length n and let A.

(i = 0, 1, . .. , n) denote the number of code words of weight i.

Then

AU, V) := l A £lr)n~l 1=0 is called the weight enumerator of C. The weight enumerators of a code C and its dual C"1 The relation is due to F. J. MacWilliams.

48

are related.

(7. 14) Theorem.

Let A(4,

77)

be the weight enumerator of an (n, k)-

code over GF(q) and let AX(4, code.

77)

bejhe weight enumerator of the dual

Then aJ"(£,

v)

= q’kA(T7 - 4,

77

+ (q - 1)4).

The proof of this theorem depends on the following lemma.

(We

write (R instead of (R^n\ ) (7. 15) Lemma.

Let

x be a non-trivial character on the group

(GF(q), +) and for any v e (R define

x

:(R“*‘C>< by

VueCR[Xv(u) := x((u’ v))]' If A is a vector space over C, f : (R -* A, and if g : (R -*• A is defined by Vue(Rtg(u)

2

f(y)x (u)l,

ve(R

then for any linear subspace Z

g(u) = IV |

Proof.

c (R and the dual subspace V-1 we have

f(v).

Z

ueV

*V

veV1 Z

Z

g(u) =

ueV

Z

ueV ve(R =

f(v)

Z

ve(R

f(v)x (u) = V Z

x((u, v)) =

ueV

= |V| Z f(v) + veV-1

f(v)

Z v£Vx

Z

x((u, v)).

ueV

In the inner sum of the second term (u, v) takes on every value in GF(q) the same number of times and since

Z

X(Q') = 0

for every

Q'eGF(q) non-trivial character, this proves the lemma. d. Remark.

/

If in (8. 5) we also require 1 to be a zero of all the code

words then the code has minimum distance at least d + 1.

The proof is

the same as above. Let us now take a look at BCH-codes from the point of view of the group theorist.

We consider a primitive BCH-code of length n = qm - 1 2

over GF(q) with prescribed zeros a, a , .. . , a

54

d— 1

(where a? is a

primitive element in GF(q

)).

bols in the code words by X.

We now denote the positions of the sym¬ (i=0, 1,

n-1) where X. := a\

extend the code by adding an overall parity check. al position by

and make the conventions

We

We denote the addition¬

1°° := 1, (a^)°° := 0 for

i ^ 0 (mod n).

We represent the code words as n-1 C +CX + ...+C ,x + c x . We shall now show that the extended o i n-1 code is invariant under the permutations of the affine permutation group on GF(q

) applied to the positions of the code.

This group consists of

the permutations

(u e GF(qm), v e GF(qm), u * 0).

PU, V(X) := uX + v

It is a doubly transitive group.

a, „0

is the cvclic J shift on the positions XQ, X^ . .. , X _r and that it leaves XM invariant.

Therefore

We first observe that P

Q transforms every extended cyclic code into itself.

Now

let (c , c , ... , cj be any code word in the extended BCH-code and apply P

to the positions to obtain (c’, c', ....7 c’on'). ' n7 o’ 11 7k = 0, 1, ... , d-1 we have v

y

i

Z C a

i

ik

v / i , \k y y /k\ Z iZ k— Z =zc.(uo! + v) = Z c. Z (,)u a v = i i 1Z=0 =

y

/k\

l

k-l

Z (,)u v 1=0 L

y

/

l \i

n

Z c (a ) = 0 i 1

because the inner sum is 0 for 1 = 0, 1, ... , d-1. muted word is also in the extended code. (8. 7)

Theorem.

n + 1 = q

m

Then for

Therefore the per¬

We have proved:

Every extended primitive BCH-code of length

over GF(q) is invariant under the affine permutation group

on GF(qm). As an application consider a primitive binary BCH-code C of n length n. Let Z a.^rj11 1 be the weight enumerator of C and let n+1 . .i=0 1 Z A. | i be the weight enumerator of C. We have i=0 1 A„. = a„. , + aand A_. , = 0. In C there are a,.. . words of 2i 2i-l 2i 2i-l 2i-l weight 2i with a 1 in position X . The total weight of the code words

55

of weight 2i in C is 2iA^.

Since the extended code is invariant under

a transitive permutation group the weight is equally divided among all the positions, i. e.

(n + l)a,,. ^ = 2iA^..

(8. 8)

The minimum weight of a primitive binary BCH-code

Theorem.

A consequence is:

is odd. Remark.

Notice that the result of Theorem 8. 8 holds for any

binary code for which the extended code is invariant under a transitive permutation group.

56

9 - Threshold decoding

There are several codes which allow a very simple error correc¬ tion procedure of which the simplest example is majority decoding.

We

consider a linear code C. (9. 1)

Definition.

An orthogonal check set of order r on a position i

of C is a set of r words of C-1, say y^, (i) (it)

= 1

y(r), such that

(v = % 2, . .. , r),

if j ^ i then yj ^ ¥= 0 for at most one value of v.

To show how decoding works let i = 1 and let a be a word with t < [|r] errors. position 1.

(a, y

Take an orthogonal check set y^\

y^

on

Then values of v if the symbol a

(v).

is correct, i

) # 0 for

r - (t-1) values of v if a '

l

’

is in error.

Since r - (t - 1) > t, the majority of the values of (a, y'^ ) (zero or not zero) decides whether a.|

is correct or not.

error can immediately be corrected. position of a code.

In the case of GF(2) the

The procedure is applied to every

Here we see one advantage of cyclic codes because

the procedure has to be applied to one fixed position only, followed by a cyclic shift of the received word etc. It is not difficult to generalise (9. 1). place 'at most one' by 'at most A'.

Suppose that in (ii) we re-

r + x - 1

Then, if there are t < [ —^—-—]

errors in a, we can use the set of parity checks, now called a A-check set of order r, to correct position i.

In the same way as above it is

easily checked that if the number of parity checks which do not yield 0 exceeds the threshold r - A(t - 1) - \ then position i is in error. Of course the reader has seen the connection to combinatorial designs.

Let A be the incidence matrix of a BIBD with parameters

57

(b, v, r, k, A). code CX

We consider the rows of A as basis vectors of a linear

over GF(q).

Then the dual C has the rows of C-1

as parity

checks and for every position i of C we can find a A-check set of order r on position i (by the definition of BIBD).

Here we encounter a problem

in coding theory which has been solved for a few cases only, namely the problem of determining the dimension of C in the situation described above.

This amounts to finding the rank of A over GF(q).

If this

rank is v, then C consists of 0 only which is hardly interesting! shall return to this problem in Chapter 11. now with q = 2.

10

110

V.

We consider one example

Let A be the incidence matrix of PG(2, 2), 10

A

We

1

0 110 0 0 11 0 0 0 1 10 0 0 0 10 0

0 0 1 0 1 1 0

0 0 0 1 0 1 1

In the example following (8. 4) we learned that the first 4 rows of A generate the cyclic code C"1 generator polynomial.

over GF(2) which has g(x) = 1 + x + x3

as

This implies that C, in reversed order, is

generated by (1 + x)(l + x

+ x ).

From our second example in Chapter

8 we know that the code C consists of the words of even weight in the (7, 4) Hamming code.

For every position of this (cyclic!) code there

are 3 rows of A which form an orthogonal check set of order 3 on that position (correcting 1 error).

Of course, there is a much simpler error

correction procedure for Hamming codes but this example serves as an introduction to the discussion of codes generated by projective planes. A further generalisation should be mentioned here.

Instead of

using a number of parity checks to check one position we can use the same procedure to determine whether a certain subset of the positions of a word contains an error.

Once this has been decided it may be

possible to use other parity checks (in the same fashion) to localise the error.

This is called multistep majority decoding (see Chapter 10). We refer the reader who is interested in learning more about the

topic of this chapter to Goethals [28] and Massey [50].

58

10 • Reed-Muller codes

This class of codes connected with finite geometries was first treated by D. E. Muller and I. S. Reed. first need a lot of notation.

To be able to introduce them we

We consider AG(m, 2),

The standard basis

vectors will be called u

u (considered as column vectors). l' ' m We make a list of all the points of AG(m, 2) as columns of a matrix E (m rows, n := 2m columns).

The order of the columns is given as

follows: column j is the binary representation of the integer j (j = 0, 1, ... , 2m-l). m i-1 If j = l L.2

(a notation we use throughout this section), then m the j-th column of E is x. := Z £..u.. We need the following definition: i=l

1]

]

(10. 1) Definition. Note that A.

A. := l

i=l

i] i

x. e AG(m, 2) | £.. = 1 i]

is an (m-1)-dimensional affine subspace (an (m-1)-

flat). (10.2) Definition.

and -

For i = 1, 2, ..., m

v := row i of E, i. e. the characteristic function of A.; i —-—i v := 1, the characteristic function of AG(m, 2). o--/

(Here the rows of E are considered as words in Or (10.3) Definition.

If a = (a , a^ ..., a^^) and b = (bQ,

\

.) > • • • ,bn_

then ab := (a b , a b . v o o’ l l1

-,b ,). ■1 n-1

Note that by (10. 1), (10. 2), and (10. 3) the product v. v. .. . v. l 2 k is the characteristic function of A. n A. n . . . n A . If the indices 11 li 12 k i i, are distinct this is an (m-k)-dimensional affine subspace of i ’ ' '' ’ k

59

AG(m, 2).

Therefore we have

(10. 4) Theorem,

w^v

.. . v^) = 2

m_ k

(10.5) Definition.

If SC {1, 2, ..., m}

{ 0, 1, ..., 2m- 1 }

is defined by

then the subset C(S) of

m i-1 i i S ^ £.. = 0 }. {]= l IJ i=l ij

C(S)

;(n) The standard basis vector e_. = (0, 0, . . . , 0, 1, 0, 0, . . . , 0) e GT (the 1 is in position j), is the characteristic function of

|Xj }.

We have

m (10. 6) e. = If {v. + (1 + £..)v }. i] o ] i=l Therefore each of the 2m basisvectors of (R^ can be wmitten as a /

polynomial of degree m-Z then LnA is empty or an affine subspace of

dimension > 0. In both cases LnA has an even number of points, i. e. 1

f.=0 if k> m-Z.

This proves that f is in the (m-Z)-th

j(i^, i2> •••» ijj)

order RM code. J Now we combine theorems (10. 9) and (10. 11). t-th order RM code of length 2m.

Let C be the

We now know that every (t+l)-flat

in AG(m, 2) provides us with a parity check equation.

These (t+l)-flats

form a BIBD with parameters: „m . = 2m-t-l (2m-l)(2m~1-l), . , (2m~t-1) v = 2 , b

,t+l

(2t+1-l)(2t-l)...(2-l)

r_(2m-l)(2m-1-l),..(2m-t-l) (2t+1-l)(2t-l). . . (2-1) Consider any t-flat T in AG(m, 2). (t+l)-flats containing T0 these (t + 1) flats.

v_. (2m-1-l)...(2m-t-l) ’

(2^1). ..(2-1)

There are 2m t - 1 distinct

Every point outside T is in exactly one of

By the procedure described in Chapter 9 we can let

a majority vote provide a parity check on the positions of T if there are less than 2m * 3

errors.

majority decoding up to 2m

By induction one sees that in t + 1 steps of 3 - 1 errors can be corrected.

Since the

minimum weight in the t-th order RM code is even we have, using (10. 4), (10. 13) Theorem. The minimum weight in the t-th order RM code of . „m . nm-t length 2 is^ 2

62

Now we take another look at groups.

By definition every code

word in the r-th order RM code of length 2m is a sum of characteristic functions of affine subspaces of dimension > m - r.

By Theorem (10. 11)

a sum of characteristic functions of affine subspaces of dimension - m - r is a code word in the r-th order RM code.

It follows that every

permutation of AG(m, 2) which also permutes the affine subspaces leaves all RM codes invariant.

This is the group of affine transformations

of AG(m, 2), a triply transitive group of order 2m "n (2m - 21). i=0 (10. 14) Theorem.

Every RM code of length 2m is invariant under all

affine transformations (acting on the positions interpreted as points in AG(m, 2)). Since the original definition of RM codes is more combinatorial we have chosen this way to present them here.

For the sake of com¬

pleteness we must mention that it was discovered later that RM codes are extended cyclic codes!

To give the cyclic definition we introduce the

notation w(j) for the weight of the binary representation of j

(interpreted

as vector over GF(2)). *

(10. 15) Theorem.

Let a be a primitive element in GF(2m).

Let

g(x) := II*(x - a^) where 11* is the product over all integers j in [0, 2m - 2] with w(j) < m - r.

If C is the cyclic code of length 2m

generated by g(x), then C is equivalent to the r-th order RM code. We do not prove this theorem here.

The proof is not difficult.

The reader interested in a proof is referred to [44],

We remark that

(10. 13) can easily be derived from (8. 6) and (10. 15).

The advantage of

the definition (10. 15) is that it can be generalised to other fields GF(q). The RM codes treated in this paragraph are examples of a larger class of codes known as Euclidean Geometry Codes (cf. [28]). Closely related to these codes are the Projective Geometry Codes. We shall give only one example. (10. 16) Definition.

oi

Let p be a prime, q = p , and let A be the incidence

matrix of points and hyperplanes of PG(m, q). a code C"1

over the alphabet GF(p).

The rows of A generate

The dual code C is called a pro-

63

jective geometry code. It can be shown (cf. e. g. [28]) that C"*" 1 + (m + P ”

)a.

An example connected to the topic of our next

chapter is the case p = a = m = 2. matrix of PG(2, 4).

The code C"1

In this case A is the incidence has dimension 10.

forward calculation one can show that C"1 weight 5.

if it is a line of PG(2, 4).

By straight¬

has exactly 21 words of

This implies that a code word of C"1

linear span!

has dimension

has weight 5 if and only

In other words: A can be retrieved from its

Once again we have an example of a linear code for which

the vectors of minimum weight (=£ 0) form a 2-design. To conclude this paragraph we give one more example linking RM-codes to a familiar part of combinatorial theory. Consider the ls^-order RM code of length 2m. (10. 10) this is the dual of the extended Hamming code.

By (10. 9) and Also by (10. 9)

this code is self-orthogonal, i. e. (a, b) = 0 for every pair of code words. The code has dimension m + 1. 1 + a is also a code-word.

For each code word a in this code

From each such pair we choose one word

and with these we form a matrix (2m rows, 2m columns).

If we re¬

place 0's by -l's in this matrix the result is a Hadamard matrix!

64

11 • Self-orthogonal codes and designs

The (8, 4)-extended binary Hamming code discussed in Chapter 7 coincides with its dual. This is an example of the class of r-th order RM 2r+l codes of length 2 (see (10. 10)) which are self-dual by Theorem (10. 9).

We now give an example of such a code over GF(3). Let

be the circulant with first row (0 1 -1 -1 1).

Define

G by 0 0 0 (11. 1) G :=

0 0 0

I

0 0

11111

s5

5

and H by ~\

H :=

1 1 1 1 1

Then GHT = 0.

S

I 5

5

J Clearly the rows of G are independent and hence G is

the generator matrix of an (11, 6) code C over GF(3) and H is a parity check matrix for C.

We remark that

r

! o o o o o o

\

I0 (11.2) GGT=!° 0 0 C

-J5

J

This implies that if a = (a^, a^, a^, a^, a^, a&) then

65

aGGTaT = -( I a.)* 2 * 1, i=2 1 i. e. all words in C

f 1 (mod 3).

have weight

Clearly no code word

can have weight 2 and in fact it is immediately clear that a linear com¬ bination of 2 rows of G produces a word of weight 5 or 6, and also that a linear combination of 3 rows has weight > 3, and since this weight cannot be 4 it must also be at least 5.

Trivially a linear combination of

more than 3 rows of G has weight > 4

and hence

— 5.

This implies

that the minimum weight and minimum distance are 5, i. e. the code is 2-error-correcting. an S(x, 2)

is

Since the number of words is 36 and the volume of

1 + 2(1]L) + 4(121) = 35, the code is perfect!

known as the ternary Golay code.

This code is

The weight enumerator is easily found

to be (11. 3)

r/11 + 132 £5 7} 6 + 132 ^6 77 5 + 330|87?3 + 110^9 772 + 24£31.

Now consider the extended code

C, a (12, 6)-code over

generator matrix G obtained from (0

-1

C

is self-dual.

(11. 4)

-1

-1

7712

-1

-1)^.

It has a

G in (11. 1) by adding a column

By (11. 2) we then have

GG^ = 0, i. e. the code

From (11. 3) we find the weight enumerator to be

+ 264776 + 440

Remark.

GF(3).

T] 3 + 24£12.

If we had started with C

instead of

C, the proof would

have been even more simple (cf. Chapter 13). Consider two code words a and b in C, both with weight 6.

Let

there be k positions where

a and b both have non-zero coordinates. k+1 Then either a + b or a - b has weight < 12 - k - [—^—]. This implies

that either

a = ±b or

k < 4.

supports a code word a in C positions of D.

(11. 5)

{1, 2, ... , 12 }

if the non-zero coordinates occur in the

Theorem.

Let

2)

be the collection of 6-subsets of the set

which support code words of weight 6 in C.

is a 5-design with parameters

66

of

We now have the following theorem.

S:= {l, 2, . . . , 12 ) 2)

We say that a subset D

Then

(12, 6, 1), i. e. a Steiner system.

Proof.

We have seen above that two code words a and b are

supported by sets which intersect in at most 4 points unless a = ±b. Therefore (11. 4) implies that there are j x 264 x 6 = 792 distinct 5-subsets of S contained in sets of the collection 5). 12 ( 5) = 792 5-subsets the proof is complete.

Since S has

Of course this is a well-known 5-design (cf. Chapter 1).

We

mention it here because it motivated a search for 5-designs of this type. In Chapter 13 we return to this question.

The result of Theorem 11. 5 is

a special case of Theorem 13. 13. We shall now show that a projective plane generates a self-dual code. 9.

A first example of a code generated by a plane was given in Chapter

Let A be the incidence matrix of PG(2, n).

We consider the sub-

2

space C A.

of

+n+^

over GF(2) which is generated by the rows of

If n is odd it is easy to see what the code C is.

If we take the sum

of the rows of A which have a 1 in a fixed position the result is a row with a 0 in that position and l's elsewhere.

These vectors generate the

subspace of (R^n +n+^ consisting of all words of even weight and this *

is C

(because C

obviously has no words of odd weight).

The case of

even n is more difficult. (11. 6) Theorem. A of

PG(2, n) generate a code C with dimension i(n2 + n + 2). Proof.

C c

If n = 2 (mod 4) then the rows of the incidence matrix

(c)

(i) Since n is even the code C is self-orthogonal, i. e.

.

Therefore dim C ^ |(n2 + n + 2).

(ii)

Let dim C = r and let k := n

2

_L

+ n + 1 - r = dim C .

Let

H be a parity check matrix of rank k for C and assume the coordinate places have been permuted in such a way that H has the form (1^ P). Tk Define N := (q T

P j ). r

Interpret A and N as rational matrices. A (

2-l

)

Then det AN1 = det A = (n + l)n2 n n . Since all entries in the first k T ki columns of AN are even we find that 2 | det A. It then follows that r > |(n2 + n + 2). From (i) and (ii) the theorem follows. J

67

From (11. 6) we immediately have: (11. 7) Theorem.

If n = 2 (mod 4) then the rows of the incidence matrix

A of PG(2, n) generate a code C, for which C is self-dual. We continue with a few other properties of the plane PG(2, n), where n = 2 (mod 4), interpreted in coding terms. (11. 8) Theorem.

The code C of Theorem (11. 6) has minimum weight

n + 1 and every vector of minimum weight is a line in PG(2, n). Proof.

Let v ^ 0 be a code word with w(v) = d.

Since every

line has a 1 as overall parity check we see that: (i)

if d is odd then v meets every line at least once and

(ii)

if d is even then every line through a fixed point of v

meets v in a second point. In case (ii) we immediately have d > n + 1. (n + l)d > n2 + n + 1, i. e. line

d ^ n + 1.

In case (i) we find

If w(v) = n + 1 then there is a

l of PG(2, n) which meets v in at least 3 points. If there is a l not on v, then every line # l through this point meets v by This would yield d > n + 3, J

point of (i).

(11, 9) Definition.

An s-arc in PG(2, n) is a set of s points no three

of which are collinear. (11. 10) Theorem.

The vectors of weight n + 2 m C are precisely the

(n+2)-arcs of PG(2, n). Proof.

(i) Let v e C and w(v) = n + 2.

l be a line and suppose v and l

in an even number of points.

Let

have 2a points in common.

Each of the n lines

these 2a points meets v at least once more. 2a + n < n + 2, i. e. (ii)

Every line meets v

*l

through one of

Therefore

a = 0 or a = 1.

Let v be an (n+2)-arc.

Let S be the set of | (n + l)(n + 2)

distinct lines of PG(2, n) through the pairs of points of v. S contains n - 1 points not in v. points.

68

Each line of

In this way we count |(n + 2)(n2 - 1)

There are n" - 1 points not in v and each of these is on at

least 2

2

(n + 2) lines of S.

(n + 2) lines of S.

Therefore each of these is on exactly

Each point of v is on n + 1 lines of S.

There¬

fore v is the sum of the lines of S, i. e. v e C. J (Also see [3],) Remark.

We leave it as an interesting exercise for the reader

to check that Theorem (11. 10) also holds for the code generated by PG(2, 4) which was treated at the end of Chapter 10. For a self-dual (n, k) code C over GF(q) we find from Theorem (7. 14) the following relation for the weight enumerator A(|, (11. 11) A(£,

77)

= q"kA(r? - 0

where k = f n.

77

+ (q - 1)0,

This means that the polynomial is invariant under the

linear transformation with matrix q~ words of C have even weight, i. e. formation £-*•-£. £> .

77):

2

(”

).

A(£,

77)

If q = 2, all code

is invariant under the trans¬

The 2 transformations generate the dihedral group

It was shown by A. M. Gleason (cf. [8], [27], [49]), that the ring

of polynomials in

£ and

free ring generated by

£

77

+

which are invariant under this group is the 77

and £

77

(£

-

) .

77

Let us now consider a possible plane of order 10.

From (11. 6),

(11. 7) and (11. 8) we know that the incidence matrix of this plane generates a code C for which C is a (112, 56)-self-dual code and that the weight enumerator A(£, ri) ’

A^ ^ = 111.

of C has coefficients A _

0

= 1, A =A =.. . =A ’12

10

=0, ’

Since all the generators of C have weight 12 and any two

words of C have an even number of l's in common we see that all weights in C are divisible by 4.

Therefore A

= A

= 0.

By sub-

stituting this in (11. 11) or by using Gleason's results one sees that

A(0

77)

is uniquely determined if we know A^, A

and A^&.

Recently

F. J. MacWilliams, N. J. A. Sloane and J. G. Thompson (cf. [48]) investigated the code words of weight 15 in C.

It was assumed that the

incidence matrix of the plane led to a code for which A^

=£ 0.

Arguments

of the same type as the ones we used in proving Theorems (11. 8) and (11. 10) severely restricted the possibilities.

In fact it was found that the

plane had to contain a particular configuration of 15 lines.

A computer

69

search showed that starting from such a configuration the plane could not be completed.

Therefore we now know that if a plane of order 10 exists,

then (11. 12) A

= 0.

In a very recent paper it was pointed out by A. Bruen and J. C. Fisher (cf. [14]) that this same result also follows from a computer result of R. H. F. Denniston concerning 6-arcs which are not contained in 7-arcs (applied to the dual plane). The idea of orthogonality can also be defined for t-designs.

A

t-design is called self-orthogonal if every two blocks intersect in an even number of points. tems.

We now consider possible self-orthogonal Steiner sys¬

Using the well-known relations between t, b, v, r, k (cf. Chapter

1) and simple counting arguments it is easy to show (an exercise for the reader; cf. [3], [6]) that there are four possibilities: (i) i. e.

t = 3, k = 4, v = 8.

This is the example of Theorem 7. 12,

AG(3, 2). (ii)

t = 3, k = 6, v = 22.

This is the unique extension of

PG(2, 4) (cf. Chapter 1 and the example following (10. 16)). (iii)

t = 5, k = 8, v = 24 (cf. Chapter 1).

(iv)

t = 3, k = 12, v = 112.

This would be the extension of a

projective plane of order 10 (if it exists). Remark.

In this way (1. 11) has been proved again.

The result (11. 12) uniquely determines the weight enumerator of the code C if we assume that the plane of order 10 is extendable, i. e. that example (iv) is possible. Further reference: [57].

70

12 ■ Quadratic residue codes

Before we can start with the actual topic of this chapter we must prove a theorem on cyclic codes (cf. Chapter 8).

v. to denote the permutation of

We use the symbol

{0, 1, . . . , n-1 }

given by w.(k) := jk (n) (mod n) and also to denote the automorphism of Or given by k ik n 7K(x ) := xJ mod (x - 1), where we use the identification of vectors in (R (n) and polynomials mod (x11 - 1).

We consider binary cyclic codes of

length n, where n is odd. (12. 1) Theorem.

For every ideal V in (R^ there is a unique poly¬

nomial c(x) e V, called the idempotent of V, with the following proper¬ ties: (i)

c (x) = c2 (x),

(ii)

c(x) generates V,

(iii)

Vf(x)cV[c(x)f(x) = f(x)], Le.

(iv)

if

Proof.

(j, n) = 1

,

then

7r.(c(x))

J

c(x) is a unit for V, is the idempotent of

7T.V.

J

(i) Let g(x) be the generator of V and let

g(x)h(x) = xn - 1

(in GF(2)[x]).

we have (g(x), h(x)) = 1.

Since x11 - 1 has no multiple zeros

Therefore there are polynomials p^x) and

p^(x) such that (12.2) px(x)g(x) + p2(x)h(x) = 1

(in GF(2)[x]).

We multiply both sides of this equation by c(x), where c(x) := pi(x)g(x). We find c2(x) + pi(x)p2(x)g(x)h(x) = c(x). Since in (R^ we have g(x)h(x) = 0, we have proved (i). (ii) by c(x) is

The monic polynomial of lowest degree in the ideal generated (c(x), x11 - 1) = (p^x)g(x), g(x)h(x)) = g(x).

71

(iii)

By (ii) every f(x) e V is a multiple of c(x).

f(x) = c(x)fi(x).

Let

Then c(x)f(x) = c2(x)f (x) = c(x)fi(x) = f(x), i. e.

c(x)

is a unit in V. (iv)

(n)

is an automorphism of GV

Since 7r

the polynomial

7r.(c(x)) is an idempotent and also a unit in n.V. 1

^

/

Since the unit is unique we have proved (iv). 7

(12. 3) Example.

We consider an example in detail.

Let n = 2m - 1.

Let m^x) be the minimal polynomial of a primitive element a of GF(2m).

Then by Theorem 8. 4 m^x) generates the Hamming code

which we now denote by H

.

Let xn - 1 = m(x)h(x) in GF(2)[x],

know that h(x) generates the dual code H*

of Hm (cf. Chapter 8).

We

This

has dimension m and hence it consists of 0 and the n cyclic

shifts of the word h(x).

It follows that there is an exponent s such that

g

f(x) := x h(x)

is the idempotent of H*^

and hence (12. 2) now has the form

Pi(x)mi(x) + xSh(x) = 1.

Clearly, the polynomial 1 + f(x) is the idempotent of H

.

We leave it

to the reader to check that f(x) has weight 2m_'*‘ (cf. final example in Chapter 10).

E. g.

if m = 4, n = 15 and m (x) = x4 + x + 1 then

(x8 + x2 + l)m (x) + x(xi:L + x8 + x7 + x^ + x' + x2 + x + 1) = 1, and xh(x) = f(x) = (x + x2 + x4 + x8) + (x3 + x6 + x9 + x12). We shall need example (12. 3) and proposition (12. 4) below in Chapter 14. By Theorem (12. 1) (ii) we have (12. 4) Proposition. q(x) {1 + f(x) )

For every polynomial q(x) the word

is in H

.

In the description of quadratic residue codes we shall use the following notation. (12. 5) Notation. (ii)

72

(i) n is an odd prime,

a is a primitive n-th root of unity in an extension field of

GF(q)

(a suitably chosen later on), (iii)

Rq

denotes the set of quadratic residues mod n,

denotes

the set of nonzero elements of GF(n)\R , o’

(iy) Xn

^ (x - a ), r eR o

gQ(x)

g (x) :=

11 (x - ar) r eR

(note that

- 1 = (x - l)g0(x)gi(x)). We assume that q^n

^^sl (mod n), i. e.

q is a quadratic residue

mod n. The nonzero elements of GF(n) are powers of a primitive element a e GF(n).

Clearly a

is a quadratic residue if e = 0 (mod 2) and a non¬

residue if e = 1 (mod 2).

Since q is a quadratic residue the set R^

is

closed under multiplication by q (mod n). (12. 6) Definition.

The cyclic codes of length n over GF(q) with

generators gQ(x) and (x - l)gQ(x) are both called quadratic residue codes (QR codes).

The extended quadratic residue code of length n + 1

is obtained by annexing an overall parity check to the code with generator «o(x)(Cf. remark following (12. 10). ) In the binary case the code with generator (x - l)g (x) consists of the words of even weight in the code with generator gQ(x).

If G is a

generator matrix for the first of these codes, then (1 1 • • 1) Lr

is a genera-

tor matrix for the second code and the generator matrix for the extended '1 11 ... 1^ 0 (cf. Chapter 7). Note that in the binary case the code is condition that

q is a quadratic residue mod n is satisfied if

n = ±1 (mod 8).

If j e R^

then the permutation Vj maps Rq into R^

and vice versa.

It is easily seen that if we replace R

by R^

in (1. 6)

we obtain equivalent codes in the sense of Chapter 7 (cf. [44] Theorem 3. 2. 2).

If n = -l (mod 4) then -1 e R^

and therefore the transformation

x -*-x-1

maps a code word of the code with generator gQ(x) into a code

word of the code with generator g^ (x).

73

n-1 If c = c(x) = Z c.x1 is a code word of the QR code n-1 i=0 1 with generator g (x), if Z c. =7 0, and if w(c) = d is the weight of this 0 i=0 1 code word, then (12. 7) Theorem.

(i)

d2 > n,

(ii)

ii n = -1 (mod 4) then d2 - d + 1 s n,

(iii)

if n = -1 (mod 8) and q = 2 then d = 3 (mod 4). n-1

Proof. by x - 1.

(i) Since

l c. * 0 the polynomial c(x) is not divisible i=0 1 By a suitable permutation 77we can transform c(x) into a

polynomial c(x) which is divisible by g (x) and of course again not * 2 n-1 divisible by x - 1. This implies that c(x)c(x) = 1 + x + x +... +x 2

The number of nonzero coefficients of the left-hand side is at most d . This proves (i). (ii) Now 77. can be chosen such that c(x) = c(x 1) and then y. 1 2 c(x)c(x) has at most d - d + 1 nonzero coefficients. d e. d -e. (iii) Let c(x) = Z x , c(x) = Z x . If e. - e = e, - e„ then i=l i=l 1 ] e. - e. = e, - e, . Hence, if any terms cancel, then four at a time, i. e. j i l k ’ ’ n = d2-d + l-4a for some a > 0. This proves (iii). ^ Now, again consider binary QR codes. (12. 8)

0(x) :=

Z xr. r eR o (n)

Clearly 0(x) is an idempotent in 6v Hence

{ 6(a) }2 = 6(a), i. e.

6(al) = 6(a) a

if i e R

nomial

6(x)

In the same way we have

and 0(o'1) + 6(a) = 1 if i e R .

and finally 0(0°) =^2^.

(12. 9) Theorem.

(2 is a quadratic residue mod n).

6(a) e GF(2).

in such a way that 0(a) = 0.

i eRj

We define

We now choose

Then 6(a1) = 0 if i e R , 0(a1) = 1 This proves:

If a in (12, 5) (ii) is suitably chosen then the poly¬

of (12. 8) is the idempotent of the QR code with generator

(x - l)gQ(x) if n = 1 (mod 8) and of the QR code with generator gQ(x) if n = -1 (mod 8). Let C be the circulant with the code word 0 as its first row. Define: 74

if

n = 1 (mod 8), n = -1 (mod 8),

c

and

It now follows from (12. 9) that the rows of G (which are not independent!) generate the extended binary QR code of length n + 1. Let us number the coordinate places of the code words in the ex¬ tended binary QR code using the coordinates of the projective line, i. e. °°, 0, 1,

n-1.

The position of the overall parity check is

°°.

We

make the usual conventions about arithmetic operations: ±0-1 = °°, ±o°

=

0,

°°

+ a =

00

for a e GF(n).

We now consider the permutations

of PSI (2, n) acting on the positions and show that the extended QR code remains invariant.

The group is generated by the transformations

Sx := x + 1 and Tx := -x 1.

Since S leaves

00

invariant and it is a

cyclic shift on the other positions it leaves the code invariant.

It is a

simple exercise to show that T maps the rows of G into linear com¬ binations of at most 3 rows of G (cf. [44] p. 83).

To most readers this

is probably a familiar property of G (compare with automorphism groups of certain Hadamard matrices). (12. 10) Theorem.

We have shown:

The automorphism group of the extended binary QR

code of length n + 1 contains PSL(2, n). Remark.

Gleason and Prange (for a discussion cf. [2]) slightly

altered the definition of extended code for the non-binary case.

They

require that the coordinate in the extended position is multiplied by a factor in such a way that the resulting code is self-orthogonal (n = -1 (mod 4)) or orthogonal to the other QR code, if extended (n = 1 (mod 4)).

They show that (12. 10) is also true for the q-ary codes.

Again, we consider an extended binary QR code. code is invariant under a doubly transitive group.

By (12, 10) this

By the remark following

(8. 8) we see that the binary QR code has odd minimum weight. allows us to apply Theorem (12. 7).

This

Thus we find:

75

(12. 11) Theorem.

The minimum weight of the binary QR code of length n

with generator g (x) is an odd number d for which (i)

d2 > n

(ii)

d2 - d + 1 2: n if ns_i (mod 8).

Remark.

if n =

1 (mod 8),

The reader familiar with the theory of difference sets

and designs should have no difficulty checking that the proof of (12. 7) (ii) and (iii) implies that, if equality holds in (12. 11) (ii), then there exists a projective plane of order d - 1

(cf. [67]).

A more combinatorial form of the square root bound (which is the name by which (12. 11) is known) was recently found by Assmus, Mattson, and Sacher (cf. [57]).

It states that if C is an (n, ^y^1) cyclic code

with minimum weight d and C

_L

^

£ C, and if the subsets of the positions,

which support code words of minimum weight, form a 2-design, then d

2

- d + 1 > n with equality if and only if the design is a projective plane

of order d - 1. We consider,a very important example of QR codes. (12. 12) Example. X23

Over GF(2) we have

- 1 = (x-l)(x11+x9+x7+x6 + x5+x+l)(x11+X1

°+X6+XJ+X4+X2

+ l) =

= (x-l)g{J(x)gi(x). The binary QR code C of length 23 with generator g (x) is a (23, 12)code.

By (12. 11) the minimum distance d of this code satisfies

d(d - 1) > 22.

Since d is odd we have d > 7.

In this case the volume

of S(x, 3) is 211, i. e. code is perfect! code.

| U S(x, e) | = 22 3 or in other words: the xeC (Cf. (7. 5). ) This code is known as the binary Golary

In this case the automorphism group of C is M2 . If n = -1 (mod 4) then the extended binary QR code C

n + 1 is self-dual.

of length

This follows from the remark preceeding (12. 7)

and (8. 3) etc. using the form we found above for the generator matrix of C.

It can also be seen from the matrix G, defined after (12. 9). We now come to one of the most important theorems of this course.

The theorem, which shows that many codes can be used to yield t-designs, is due to E. F. Assmus, Jr. , and H. F. Mattson, Jr. (cf. [2]).

76

(12. 13) Theorem.

Let A be an (n. k)-eode over GF(q) and let AX be

the (n, n - k) dual code. an(t e.

Let the minimum weights of these codes be d

Let t be an integer less than d.

satisfying vq - [

Let v

be the largest integer

v0+q-2 ^—1 < d, and wq the largest integer satisfying

w +q-2 wo - [——j—] < e, where, if q = 2, we take vq = wq = n.

Suppose

the number of non-0 weights of A± which are less than or equal to n - t is itself less than or equal to d - t.

Then, for each weight v with

d < v < v , the subsets of S := {1, 2, . .. , n ) of weight d in A form a t-design. with e < w < min {n - t, w of weight w in A Proof. we write B.

_L

which support code words

Furthermore, for each weight w

}, the subsets of S which support code words 0

form a t-design.

In the proof we shall use the following notation.

For A±

If T is a fixed t-subset of S and C is any code then we

denote by C' the code obtained by deleting the coordinates in T from the code words of C.

We denote by C

the subcode of C which consists of o the words of C which have zeros at all the positions of T. The proof is in 6 steps. (i)

By definition of

two words of A with weight < vq which

have the same support must be scalar multiples of each other (because there exists a linear combination of these two words with weight < d). An analogous statement holds for B. (ii)

We shall use the following property of t-designs.

be a t-design and let T be a t-subset of S. blocks D of 2) with t

,

|d fl t| = k.

Then by (1. 1) we have

K

J

(3 = 0, 1,

t).

j

By solving these equations we find that of the set T.

the number of

.

I (V = (X

k=0 J

Denote by

Let (S, 2))

does not depend on the choice

The statement for k = 0 implies that the complements of

the blocks of 2) also form a t-design. (iii)

Consider any d - 1 columns of the generator matrix of A

and delete these.

From the definition of d it follows that the resulting

matrix still has rank k.

77

(iv) Consider any t-subset T of S. code.

Clearly (B )' c (A’)"*".

By (iii) A'

is an (n-t, k)-

The dimension of B'q is at least n - k - t.

Hence B' = (A’) . Let 0 2

Hence the minimum distance is

Therefore the extended code C has minimum distance d = 12 and

again it is self-dual.

One can compute the weight enumerator of this code

using Theorem (7. 14).

The only occurring weights are 0, 12, 16, 20,

24, 28, 32, 36 and 48.

Hence (12. 13) applies with t = 5.

v = 12, 16, 20 or 24.

This yields 4 different 5-designs.

We can take If we take

v = 28, 32, 36 we find the complementary designs. These examples explain the interest (from the point of view of design theorists) in determining the minimum weight of quadratic residue codes.

Very little is known in this area.

A survey for n < 59 can be

79

found in [4]. It would be beyond the scope of this course to go into a method of treating QR codes which looks very promising. contractions of self-orthogonal codes.

This is the idea of

The idea is to use an element of

the automorphism group of a self-orthogonal code to map the code into a code of smaller length which is still self-orthogonal.

This method was

used successfully by Assmus and Mattson to determine the minimum weight of the (60, 30) extended QR code over GF(3) minimum weight turned out to be 18. (12. 13) is satisfied if t = 5.

(cf. [5]).

The

This means that the condition of

Hence this QR code yields 5-designs on 30

points. We give one more example of (12. 13) which will turn up again later. (12. 19) Example.

„

.,

,„

,

2Z-1 n 22-1 Consider a (2 - 1, 2

correcting primitive binary BCH-code C

1-21 )-2-error-

(cf. (8. 5)).

Using Theorem

(7. 14) one can show that in C the only weights which occur are 22Z-2 ± 2l-1 _ _ 21-1 ,22-1 Now take n 2 , k = 2 - 1 - 21, q = 2

22l-2

_

and let A be C.

J_

In A^

only 3 weights occur.

t = 3 the condition of (12. 13) is satisfied.

Hence if we take

It follows that for each v

the subsets which support code words of weight v form a 3-design. Further references for this chapter:

80

[49], [641, [70].

13 • Symmetry codes over GF(3)

In this section we treat a sequence of codes which have a number of properties in common with QR codes, e. g. they also lead to new 5designs.

The results are due to V. Pless ([531, [54]).

Definition.

(13. 1)

A C-matrix of order

m

with entries T ±1 (outside the diagonal) and 0 (on the diagonal) such that CC =(m-l)I . A lot is known about these matrices.

is a matrix C

The first construction is the

well known Paley construction: Let q

be a power of an odd prime and let

x

be the quadratic

character on

GF(q).

order

using the coordinates of the projective line of order q,

i. e.

q + 1

We number the rows and columns of a matrix of

°° and the elements of GF(q).

c

, := x(b - a) ci, 0

Define

by

(a, b e GF(q)).

Then C

, , is a C-matrix of order q + 1 (cf. [381). We note that q+1 C ,CT , = -I ,, over GF(3) if q = -1 (mod 3) and that C ,, is q+1. q i-l q+1 q ■ i symmetric if q = 1 (mod 4), skew-symmetric if q = -1 (mod 4).

(13. 2)

Definition.

dimension q + 1

Let q = -1 (mod 6). as a

We define a symmetry code of

(?q + 2, q + l)-code

Sym^

+2

ovcr

GP'(3) with

If q

is a power of an odd

generator

°2q+2 where

(Iq+i Cq+1)’ is a C-matrix of order

prime then we take for

C +;,

q+1.

the Paley matrix defined above.

We prove a number of properties of these codes:

81

Proposition.

(13.3)

A

Sym2q + 2

is self-dual (and hence all weights

are divisible by 3).

Proof.

This follows from Cq+^Cq+1 = -Iq+1

(13.4) Proposition. matrix for

Sym

Pr00f.

G*q+2 := ((-l)(q+1)/2C_^ I_^) 'q+1 q+1

7

0

I

(.l/q+l^I q+1

Proof.

The linear transformation of (R^2ci+2^

q+1

Sym2q+2

invariant.

This follows from (13. 4). 1 the Golay codes are the only nontrivial perfect codes if the size of the alphabet is a power of a prime (cf. [45], [66]). We now present some of the theory of binary nearly perfect codes due to J. -M. Goethals and S. L. Snover (cf. [31]).

These codes are a

special case of the class of uniformly packed codes, introduced by N. V. Semakov, V. A. Zinovjev and G. V. Zaitzef (cf. [60]). codes also lead to t-designs.

These

Necessary conditions for the existence of

such codes are very similar to those for perfect codes.

As yet, these

have not led to non-existence proofs except for t < 5 (cf. [45]).

In this

chapter all codes are binary. Let C c t for all w e Nd(v). subsets of

{1, 2, . .. , n }

to the design 25 . design.

{1, 2,

Proof. (k fixed).

which are not contained in a subset belonging form a t-

is the t-design formed by all the other (t + 1)n).

(14.9) Theorem. with parameters

contains all the (t + 1)-

The (t + l)-subsets of the blocks of 25 x

Apparently 252

subsets of

Hence 25^

J

The design 25^

can be extended to a (t + l)-design

(n+1, d+1, A = [(n-t)/(t+l)]. Consider C and delete the k-th coordinate of each word

The resulting code Ck again has word length n, minimum

distance d and

IC, I = |c|. Hence C, is nearly perfect. Since this is 1 k k true for every k it follows from Theorem (14. 7) that the collection of (d + 1)-subsets of

{1, 2, ..., n, n+1 }

which support the words u - v,

where u eC, v e C,' u e Nd+] (v), form a (t + l)-design with the same A as in Theorem (14. 7). By now, the more pessimistic design-theorists begin to wonder whether there are any nearly perfect codes! (14. 10) Example.

Consider the parity check matrix of a binary Hamming

code (cf. Chapter 7) and delete any column.

We obtain the parity check

matrix of a linear code with d = 3, n = 2m - 2 and dimension 2m - 2 - m. We have equality in (14. 3), i. e. this code is nearly perfect, but not perfect. The Preparata Codes (cf. [551).

We now introduce a sequence of codes

which are of great combinatorial interest. tion.

Let m > 3.

We need quite a lot of prepara¬

Let H^ be the cyclic Hamming code of word length

n = 2m - 1 as described in (12. 3).

The word 1 considered as a poly¬

nomial in GF(2) [x] (mod xn - 1) is u(x) := (x11 - l)/(x - 1). be the cyclic code with generator (x - l)m (x)m (x).

Let

This BCH-code is

a subcode of H^ and by the remark following Theorem (8. 6) we see that this code has minimum distance at least 6. We define the code C

90

m

by

(14. 11) Cm := {(m(x), i, m(x) + (m(l)+i)u(x)+s(x)) | m(x)eHTvi, ie {0, 1 ), s(x)eB m m This is obviously a linear code with word length

2n + 1

and dimension

equal to) dim H

+ 1 -+ dim B = 2n - 3m. m mm We now prove:

(14. 12) Lemma,

Proof. (i)

C

Since the code is linear we can consider weights.

Let

. m (ii)

has minimum distance > 5. m -

m(x) = 0, i = 0, s(x) 4 0.

Then w(s(x)) > 6 because

s(x) eB

Let

m(x) = 0, i = 1. 3

3

u(o) + s(ce) = u(o' ) + s(g ) = 0

Then u(x) + s(x) 4 0,

and therefore w(u(x) + s(x)) > 5 by

Theorem (8. 6). (iii)

m(x) 4 0.

finished unless

Since

m(x) + (m(l) + i)u(x) + s(x) e Hm we are

m(x) +(m(l) + i)u(x) + s(x) = 0.

substituting x = a

that

Now consider

m(a ) = 0, i. e.

S(0, 1)

(R

(2n+l)

w(m(x)) > 5.

/

in polynomial form, i. e. xn_1 } c GF(2)[x] (mod x11 - 1).

S(0, 1) = {0, 1, x, x ,

In

In that case we find by

we consider the set of vectors

(14. 13) S := {(q(x), 0, q(x)f(x)) |q(x) e S(0, 1)), where

f(x)

is the idempotent of

(14. 14) Definition. C

given by m ---

C

m

defined in (12. 3).

The Preparata code

Km is the union of cosets of

+ S.

It is obvious that no 2 elements of S are in the same coset.

Hence

Ik | = (n + i)|c |. 1 m1 1 m (14. 15) Theorem.

For odd m the Preparata code

Km

is a nearly

perfect code with minimum distance 5.

Proof. two words of

(i) Km

Since

K

is nonlinear we must now consider any

and determine the weight of their sum.

By Lemma

(14. 12) it is sufficient to consider words from different cosets.

Such a

91

sum has the form (14. 16) (m(x), i, m(x) + (m(l)+i)u(x) + s(x)) + (qi (x)+q2 (x), 0, (qx (x)+qz(x))f(x)). g

Since H

m

is perfect there is a unique element x

e S(0, 1)

such that

m'(x) := qx(x) + q?(x) + xS e Hm. Observe that

m'(x) = 0

if q (x) = 0

or q?(x) = 0.

We define

m"(x) := xS(l + f(x)) + m’(x) + m'(l)u(x). We remark that

m"(x) e Hm

considered mod xn - 1 by m(x)

(cf. (12. 3)).

we have

Since all polynomials are

m’(x)f(x) = 0.

We replace

m(x) + m'(x)

and rewrite (14. 16) as

(14. 17) (m(x), i, m(x) + (m(l)+i)u(x) + s(x)) + (xS, 0, xS) + (0, 0, m"(x)). (ii)

From (14. 17) we find that the weight W which we wish to

determine is (14. 18) W = w(m(x)+xS) + i + w(m(x) + (m(l)+i)u(x) + s(x) +xS + m"(x)). s i k has the form x + xJ + x , and 3 3S h 2h by definition m’(a) = 0. It follows that m'(a ) = a (a + a ), where 3 h = j - s * 0. If, on the other hand, m'(x) = 0 then m'(a ) can also be (ii)

written as

If

m’(x) * 0, then

m'(x)

a^s(a^ + a^); where

(iv)

Since

m

h = 0.

is odd the equation

3h a' =1

has

h = 0

as its only

solution

(ah e GF(2m)). It follows that 1 + ah + a2h * 0 for all •7 3q 7 3^ h "?Y\ Therefore m"(a) = a + m’(a') = a' (1 + a + a ) # 0.

h.

s

Let $(x) be any word with 0(1) = 0, 0(a) = a , 3 3s h 2h 0(a ) = a (a + a ) for some h. Then 0(x) has weight at least 4. (v)

To prove this, first observe that 0(x) even weight.

clearly is not 0 and that it has

If 0(x)

has weight 2 this would imply that there are two m s elements x and x in GF(2 ) such that x + x = a and 1 h 2 2h 1 2-q h x3 + x3 = a^s(a3 + a ). From this we find that a x + a and 1 2 ' 1 -s h 2 a x^ + a are the solutions of the equation 1 + £ + £, =0. We showed above that this equation has no solutions in

92

GF(2

)

if

m

is odd.

(vi)

Let s(x) e Bm
4.

In the remaining part of the proof we consider 5 different possible forms of (14. 17). (vii)

Let

m(l) = 0, i = 1.

By (14. 18) we have

W > 1 + w(s"(x)),

where

s"(x) := u(x) + s(x) + m"(x).

s"(o?3) t 0

Since

by (iv), we find w(s"(x)) > 4

(viii) Let have w(m(x)) > 4.

s"(l) = s"(a) = 0

and

from Theorem (8. 6).

m(l) = 0, i = 0, m(x) # 0.

Then by Theorem (8. 6) we

Then (14. 18) yields

W > w(m(x)) + w(m(x) + m"(x) + s(x)) -2>4 + 3- 2 = 5,

once again using Theorem (8. 6) and the fact that m(l) + m"(l) + s(l)= 1 and

m(cv) + m"(a) + s(o) = 0/ (ix)

Let

m(x) = 0, i = 0.

W s 1 + w(s ’ (x)) (x) (p(x) ^ 0.

Let

Then we find from (14. 18) and (14. 19)

5.

m(l) = 1, i = 0.

Let

(x) := m(x) + u(x) + s'(x).

Then

From (14, 18) we find

W > w(m(x) + xS) + w(0(x)). g

By Theorem (8. 6) w(m(x) + x ) > 4 unless case

0(x)

m(x)

is a trinomial.

satisfies the conditions of (v) and hence w(0(x)) > 4.

In that

So in

both cases W ^ 5. (xi)

W

Let m(l) =1, i = 1.

From (14. 18) we find

w(m(x) + xS) + 1 + w(m(x) + s'(x)).

In the same way as in (x) we are finished unless m(x) = xS + xc + xd.

In that case

m(x)

is a trinomial, say

m(l) + s’(1) = 0, m(x) + s'(x) * 0,

and the result follows from Theorem (8. 6).

93

The steps (vii) to (xi) prove that Km (xii)

We have already seen that

The word length is

2n + 1

i |K

has minimum distance 5. | „m„2n-3m _ „2n-2m | =22 - 2

and d = 2t + 1 = 5.

Ik + Tn1 {1 ‘ + (2"+1) ' 1 ' +' (2no+1) v 2

If

m

is odd we have

2m (2n2+1) ^

^ } = 2'n+1

i. e. equality holds in (14. 3).