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 [21].

Association schemes were introduced by Bose and Shimamoto [13] 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, …, Γn, satisfying the conditions

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

(ii) given p, q ϵ X with {p, q} ϵ Γk the number aijk of r ϵ X with {p, r} ϵ Γi, {q, r} ϵ Γj 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.