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Wikipedia / 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, the theory of association schemes generalizes the character theory of linear representations of groups.

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. 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 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. The definition states that if x and y are ith associates then so are y and x. Every pair of points are i. Each point is its own zeroth associate while distinct points are never zeroth associates. If x and y are k which are both i and j.

Graph interpretation and adjacency matrices

A symmetric association scheme can be visualized as a complete graph is the valency of the relation.

The relations are described by their adjacency matrices and is a v × v matrix.

are v × v (0,1)-matrices which satisfy

  1. is symmetric,

  2. (the all-ones matrix),

  3. ,

  4. .

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's:

Terminology

  • 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 (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 Ph. Delsarte who recognized and fully used the connections with coding theory and design theory.

A generalization called coherent configurations has been studied by D. G. Higman.

Basic facts

  • .

  • .

The Bose–Mesner algebra

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 Hadamard (entrywise) product. The algebra formed with the matrix product is called the Bose–Mesner algebra of the association scheme.

are symmetric and commute with each other, they can be diagonalized is semi-simple and has a unique basis of primitive idempotents.

matrices which is isomorphic, and is often easier to work with.

Examples

  • The Johnson scheme, denoted by J(v, k), is defined as follows. Let S be a set with v elements. The points of the scheme J(v, k subsets of S with k elements. Two k-element subsets A, B of S are ith associates when their intersection has size k−i.

  • The Hamming scheme, denoted by H(n, q), is defined as follows. The 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 ith 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 ith associates if their distance is i.

  • A finite group G, with a class Rg 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) 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

Notes

References