Association scheme

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, ..., R_n which satisfy:

• $R\_\{0\}\; =\; \backslash \{(x,x)\; :\; x\; \backslash in\; X\backslash \}$; it is called the identity relation.
• Defining $R^*\; :=\; \backslash \{(x,y)\; :\; (y,x)\; \backslash in\; R\backslash \}$, if R in S, then R* in S.
• If $(x,y)\; \backslash in\; R\_\{k\}$, the number of $z\; \backslash in\; X$ such that $(x,z)\; \backslash in\; R\_\{i\}$ and $(z,y)\; \backslash in\; R\_\{j\}$ is a constant $p^k\_\{ij\}$ depending on $i$, $j$, $k$ but not on the particular choice of $x$ and $y$.

An association scheme is commutative if $p\_\{ij\}^k\; =\; p\_\{ji\}^k$ for all $i$, $j$ and $k$. 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 $R\_i$ is a symmetric relation. That is:

• if (x, y) ∈ R_i, then (y, x) ∈ R_i. (Or equivalently, R* = R.)

Every symmetric association scheme is commutative.

Two points x and y are called i th associates if $(x,y)\; \backslash in\; R\_i$. 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 $i$. 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 $z$ which are both i th associates of $x$ and j th associates of $y$ is a constant $p^k\_\{ij\}$.

A symmetric association scheme can be visualized as a complete graph with labeled edges. The graph has $v$ vertices, one for each point of $X$, and the edge joining vertices $x$ and $y$ is labeled $i$ if $x$ and $y$ are $i$ th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled $k$ having the other edges labeled $i$ and $j$ is a constant $p^k\_\{ij\}$, depending on $i,j,k$ but not on the choice of the base. In particular, each vertex is incident with exactly $p^0\_\{ii\}=v\_\{i\}$ edges labeled $i$; $v\_\{i\}$ is the valency of the relation $R\_\{i\}$. There are also loops labeled $0$ at each vertex $x$, corresponding to $R\_\{0\}$.

The relations are described by their adjacency matrices. $A\_i$ is the adjacency matrix of $R\_\{i\}$ for $i=0,\backslash ldots,n$ and is a v × v matrix with rows and columns labeled by the points of $X$.

:$\backslash left(\; A\_i\; \backslash right)\_\{x,y\}\; =\; \backslash begin\{cases\}$

1, & \mbox{if } (x,y) \in R_{i},\\

0, & \mbox{otherwise.} \end{cases}\qquad (1)

The definition of a symmetric association scheme is equivalent to saying that the $A\_i$ are v × v (0,1)-matrices which satisfy

:I. $A\_i$ is symmetric,

:II. $\backslash sum\_\{i=0\}^n\; A\_i\; =\; J$ (the all-ones matrix),

:III. $A\_0\; =\; I$,

:IV. $A\_i\; A\_j\; =\; \backslash sum\_\{k=0\}^n\; p^k\_\{ij\}A\_k\; =\; A\_j\; A\_i,\; i,j=0,\backslash ldots,n$.

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 $A\_\{i\}$ contain $v\_\{i\}$ $1$'s:

:$A\_\{i\}\; J=J\; A\_\{i\}=v\_\{i\}\; J.\; \backslash qquad(2)$

• The numbers $p\_\{ij\}^k$ are called the parameters of the scheme. They are also referred to as the structural constants.

The term association scheme is due to {{harv|Bose|Shimamoto|1952}} but the concept is already inherent in {{harv|Bose|Nair|1939}}.{{harvnb|Dembowski|1968|loc=pg. 281, footnote 1}} 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 {{harv|Bose|Mesner|1959}} and the introduction of the Bose–Mesner algebra. The most important contribution to the theory was the thesis of Ph. Delsarte {{harv|Delsarte|1973}} who recognized and fully used the connections with coding theory and design theory.{{harvnb|Bannai|Ito|1984|loc=pg. vii}}

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

• $p\_\{00\}^0\; =\; 1$, i.e., if $(x,y)\; \backslash in\; R\_0$ then $x\; =\; y$ and the only $z$ such that $(x,z)\; \backslash in\; R\_0$ is $z=x$.
• $\backslash sum\_\{i=0\}^\{k\}\; p\_\{ii\}^0\; =\; |X|$; this is because the $R\_i$ partition $X$.

The adjacency matrices $A\_i$ of the graphs $\backslash left(X,R\_\{i\}\backslash right)$ generate a commutative and associative algebra $\backslash mathcal\{A\}$ (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 $\backslash mathcal\{A\}$ are symmetric and commute with each other, they can be diagonalized simultaneously. Therefore, $\backslash mathcal\{A\}$ is semi-simple and has a unique basis of primitive idempotents $J\_\{0\},\backslash ldots,J\_\{n\}$.

There is another algebra of $(n+1)\backslash times(n+1)$ matrices which is isomorphic to $\backslash mathcal\{A\}$, and is often easier to work with.

• 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) are the $\{v\; \backslash choose\; k\}$ 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 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 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 $X=G$, with a class R_g for each group element, as follows: for each $g\; \backslash in\; G$ let $R\_g\; =\; \backslash \{(x,y)\; \backslash mid\; x=g*y\backslash \}$ 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:{{harvnb|Street|Street|1987|loc=pg. 238}}

: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 R_s.

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.

• Block design
• Bose–Mesner algebra
• Combinatorial design

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