Given two k-subsets A, B of an n-set, n ≥ 2k, there are k + 1 possible relations between them: they may be equal, they may intersect in k − 1 elements, they may intersect in k − 2 elements,…, or they may be disjoint.

Given two words (k-tuples) a, b ∈ Ak, where A is an ‘alphabet’ of size at least 2, there are k + 1 possible relations between them: they may be equal, they may agree in k − 1 coordinates, they may agree in k − 2 coordinates,…, or they may disagree in all coordinates.

These instances of a set together with a list of mutually exclusive and exhaustive binary relations are examples of association schemes, which we define shortly. Association schemes provide one of the foundations of combinatorics and so we include this chapter even though it will be difficult reading. They have been implicit in many of the previous chapters; we have explicitly discussed 2-class association schemes, as they are equivalent to the strongly regular graphs discussed in Chapter 21. This chapter elaborates on some of the material of Chapter 21 but has different goals.

Association schemes arose first in the statistical theory of design of experiments, but the work of Ph. Delsarte (1973) has shown how they serve to unify many aspects of our subject. In particular, certain results of coding theory and the theory of t-designs—which were originally discovered independently—are now seen to be ‘formally dual’ aspects of the same ideas in association schemes.