Q polynomial association schemes
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. Indeed, in algebraic combinatorics, association schemes provide a unified approach to many topics, for example combinatorial designs and coding theory. In algebra, association schemes generalize groups, and the theory of association schemes generalizes the character theory of linear representations of groups.
Contents
- Q polynomial association schemes
- Hypergroups association schemes buildings
- Definition
- Graph interpretation and adjacency matrices
- Terminology
- History
- Basic facts
- The BoseMesner algebra
- Examples
- Coding theory
- References
Hypergroups association schemes buildings
Definition
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:
An association scheme is commutative if
A symmetric association scheme is one in which each relation
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
Graph interpretation and adjacency matrices
An association scheme can be visualized as a complete graph with labeled edges. The graph has
The relations are described by their adjacency matrices.
The definition of an association scheme is equivalent to saying that the
The (x, y)-th entry of the left side of (IV) 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
Terminology
History
The term association scheme is due to (Bose & Shimamoto 1952) but the concept is already inherent in (Bose & Nair 1939). 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 (Bose & Mesner 1959) and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of P. Delsarte (Delsarte 1973) 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).
Basic facts
The Bose–Mesner algebra
The adjacency matrices
Since the matrices in
There is another algebra of
Examples
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.