points of H(n, q) are the qn ordered n-tuples over a set of size q. Two n-tuples x, y are said to be i th associates if they disagree in exactly i coordinates. E.g., if x = (1,0,1,1), y = (1,1,1,1), z = (0,0,1,1), then x and y are 1st associates, x and z are 1st associates and y and z are 2nd associates in H(4,2). A distance-regular graph, G, forms an association scheme by defining two vertices to be i th associates if their distance is i. A finite group G yields an association scheme on , with a class Rg for each group element, as follows: for each let where is the group operation. The class of the group identity is R0. This association scheme is commutative if and only if G is abelian. A specific 3-class association scheme: Let A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i, j&hairsp;) entry is s if elements i and j are in relation Rs. Coding theory The Hamming scheme and the Johnson scheme are of major significance in classical coding theory. In coding theory, association scheme theory is mainly concerned with the distance of a code. The linear programming method produces upper bounds for the size of a code with given minimum distance, and lower bounds for the size of a design with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain polynomial properties; this leads one into the realm of orthogonal polynomials. In particular, some universal bounds are derived for codes and designs in polynomial-type association schemes. In classical coding theory, dealing with codes in a Hamming scheme, the MacWilliams transform involves a family of orthogonal polynomials known as the Krawtchouk polynomials. These polynomials give the eigenvalues of the distance relation matrices of the Hamming scheme. See also Block design Bose–Mesner algebra Combinatorial design Notes References . (Chapters from preliminary draft are available on-line.) Design of experiments Analysis of variance Algebraic combinatorics Representation theory