After a short account of the theory of association schemes, this final chapter contains an outline of part of the thesis of P. Delsarte, in which many of the concepts of classical coding theory and design theory are generalised to classes of association schemes. For proofs, we refer the reader to [39].

Association schemes were introduced by Bose and Shimamoto [18] as a generalisation of strongly regular graphs. An association scheme consists of a set X together with a partition of the set of 2-element subsets of X into n classes Γ1,…, Γnsatisfying the conditions

(i)given p ε X, the number ni. of q ε X with {p, q}εΓ1 depends only on i;

(ii) given p, q ε X with {p, q } ε εΓk, the number of rε X with {p, r} εΓi εΓi {q, r } εΓi depends only on i, j, and k. It is convenient to take a set of n ‘colours’ c1, …, cn, and colour an edge of the complete graph on X with colour ci if it belongs to Γi; so Γi. is the ci.-coloured subgraph. The first condition asserts that each graph Γi. is regular; the second, that the number of triangles with given colouring on a given base depends only on the colouring and not on the base. A complementary pair of strongly regular graphs forms an association scheme with two classes, and conversely.

The predecessor of this book (London Mathematical Society Lecture Note Series 19) had its origin in several short courses of lectures given by the authors at Westfield College, London, in 1973. The audience for the lectures consisted mainly of design theorists, and the aim was to present developments in graph theory and coding theory having a bearing on design theory. An introductory chapter on designs was added, for the benefit of readers without the background of the Westfield audience.

For the present volume, the format has been kept, but extensive revisions and updatings have been made. New material includes ovals in symmetric designs (Chapters 1 and 13), the inequalities of Ray-Chaudhuri and Wilson (Chapter 1), partial geometries, with the Hoffman-Chang and Hall-Connor theorems (Chapter 4), 1-factorisations of K6 (Chapter 8), equidistant codes (Chapter 12), planes and biplanes (Chapter 13), generalised quadratic residue codes and inversive planes (Chapter 14), two-weight projective codes (Chapter 16), and the Krein bound (Chapter 17).

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. Γ is connected with diameter 2 if d > 0, and is disconnected if d = 0. Γ has a girth provided a > 1; the girth is 3 if 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 a2 + 1 vertices; and a graph with girth 5 and minimal valency a has at least a2 + 1 vertices. Equality holds in either case if and only if the graph is strongly regular with c = 0, d = 1. Such a graph is called a Moore graph with diameter 2. It is worth noting that analogous bounds exist for larger values of diameter and girth, but recently Bannai and Ito[11]and Damerell [35] have shown that they are attained only by graphs consisting of a single circuit.

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 possibility, that we have a set Δ 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} and {prs} are blocks, we can decide whether {qrs} is a block.

The theory of designs concerns itself with questions about subsets of a set (or relations between two sets) possessing a high degree of symmetry. 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 [16], whose definition reflects the symmetry inherent in t-designs. First, however, we shall mention an 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 consists 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 together with a symmetric irreflexive binary relation (called adjacency) on the vertex set.

The two 5-designs connected with the Golay codes have very interesting automorphism groups. It is easy to see that the collection of (2e + 1)-subsets of {1, 2,…, n } which support code words of weight 2e + 1 in a binary perfect e-error-correcting code of word length n form an (e + l)-design with parameters (n, 2e + 1, 1), i. e. a Steiner system (cf. [79] (5. 2. 6)). This design can be extended to an (e + 2)- design. These two statements explain the interest of both group- and design-theorists in perfect codes. However, it has been shown that for e > 1 the Golay codes are the only nontrivial perfect codes if the size of the alphabet is a power of a prime (cf. [80], [114]).

We now present some of the theory of binary nearly perfect codes due to J. -M. Goethals and S. L. Snover (cf. [57]). 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. [106]). These codes also lead to t-designs. Necessary conditions for the existence of such codes are very similar to those for perfect codes. The theory of uniformly packed codes was developed by J. -M. Goethals and H. C. A. van Tilborg. For a survey of most of what is known about these codes we refer to [115], Later in this chapter we shall describe some of the connections between uniformly packed codes, so-called two-weight codes and strongly regular graphs.