Association scheme

The theory of association schemes arose in statistics, in the theory of experimental design for the analysis of variance. In mathematics, association schemes belong to both algebra and combinatorics. In algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and the theory of error correcting codes. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.

An n class association scheme consists of a set X together with a partition S of X × X into n + 1 binary relations, R0, R1, , Rn which satisfy:

; it is called the identity relation. Defining , if R in S, then R* in S.
If , the number of such that and is a constant depending on , , but not on the particular choice of and
An association scheme is commutative if for all , and Most authors assume this property. Note, however, that while the notion of an association scheme generalizes the notion of a group, the notion of a commutative association scheme only generalizes the notion of a commutative group. A symmetric association scheme is one in which each is a symmetric relation. That is:

if (x, y) ∈ Ri, then (y, x) ∈ Ri. (Or equivalently, R* = R.)
Every symmetric association scheme is commutative. Two points x and y are called i th associates if The definition states that if x and y are i th associates then so are y and x. Every pair of points are i th associates for exactly one Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k th associates then the number of points which are both i th associates of and j th associates of is a constant .

The numbers are called the parameters of the scheme. They are also referred to as the structural constants.

The term association scheme is due to but the concept is already inherent in These authors were studying what statisticians have called partially balanced incomplete block designs (PBIBDs). The subject became an object of algebraic interest with the publication of and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte who recognized and fully used the connections with coding theory and design theory. Generalizations have been studied by D. G. Higman (coherent configurations) and B. Weisfeiler (distance regular graphs).

, i. e., if then and the only such that is ; this is because the partition .

The adjacency matrices of the graphs generate a commutative and associative algebra (over the real or complex numbers) both for the matrix product and the pointwise product. This associative, commutative algebra is called the Bose–Mesner algebra of the association scheme. Since the matrices in are symmetric and commute with each other, they can be diagonalized simultaneously. Therefore, is semi simple and has a unique basis of primitive idempotents
There is another algebra of matrices which is isomorphic to , and is often easier to work with.

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.