Every symmetric association scheme is commutative. 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. Two points x and y are called i th associates if. The definition states that if x and y are i th associates 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.
A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has vertices, one for each point of, and the edge joining vertices and is labeled if and are th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled having the other edges labeled and is a constant, depending on but not on the choice of the base. In particular, each vertex is incident with exactly edges labeled ; is the valency of the relation. There are also loops labeled at each vertex, corresponding to. The relations are described by their adjacency matrices. is the adjacency matrix of for and is a v × vmatrix with rows and columns labeled by the points of. The definition of a symmetric association scheme is equivalent to saying that the are v × v -matrices which satisfy The -th entry of the left side of is the number of paths of length two between x and y with labels i and j in the graph. Note that the rows and columns of contain 's:
Terminology
The numbers are called the parameters of the scheme. They are also referred to as the structural constants.
History
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. 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 and B. Weisfeiler.
Basic facts
, i.e. if then and the only such that is
, this is because the partition.
The Bose–Mesner algebra
The adjacency matrices of the graphs generate a commutative and associative algebra 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.
Examples
The Johnson scheme, denoted J, is defined as follows. Let S be a set with v elements. The points of the scheme J are the subsets of S with k elements. Two k-element subsets A, B of S are i th associates when their intersection has size k − i.
The Hamming scheme, denoted H, is defined as follows. The points of H 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 =, y =, z =, then x and y are 1st associates, x and z are 1st associates and y and z are 2nd associates in H.
A distance-regular graph, G, forms an association scheme by defining two vertices to be i th associates if their distance is i.
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.
