kb/data/en.wikipedia.org/wiki/Association_scheme-1.md

title	chunk	source	category	tags	date_saved	instance
Association scheme	2/3	https://en.wikipedia.org/wiki/Association_scheme	reference	science, encyclopedia	2026-05-05T09:48:58.210111+00:00	kb-cron

== 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 Ph. Delsarte (Delsarte 1973) 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 ==

      p
      
        00
      
      
        0
      
    
    =
    1
  

{\displaystyle p_{00}^{0}=1}

, i.e., if

    (
    x
    ,
    y
    )
    ∈
    
      R
      
        0
      
    
  

{\displaystyle (x,y)\in R_{0}}

then

    x
    =
    y
  

{\displaystyle x=y}

and the only

    z
  

{\displaystyle z}

such that

    (
    x
    ,
    z
    )
    ∈
    
      R
      
        0
      
    
  

{\displaystyle (x,z)\in R_{0}}

is

    z
    =
    x
  

{\displaystyle z=x}

.

      ∑
      
        i
        =
        0
      
      
        k
      
    
    
      p
      
        i
        i
      
      
        0
      
    
    =
    
      |
    
    X
    
      |
    
  

{\displaystyle \sum _{i=0}^{k}p_{ii}^{0}=|X|}

; this is because the

      R
      
        i
      
    
  

{\displaystyle R_{i}}

partition

    X
  

{\displaystyle X}

.

== The Bose–Mesner algebra == The adjacency matrices

      A
      
        i
      
    
  

{\displaystyle A_{i}}

of the graphs

      (
      
        X
        ,
        
          R
          
            i
          
        
      
      )
    
  

{\displaystyle \left(X,R_{i}\right)}

generate a commutative and associative algebra

        A
      
    
  

{\displaystyle {\mathcal {A}}}

(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. Since the matrices in

        A
      
    
  

{\displaystyle {\mathcal {A}}}

are symmetric and commute with each other, they can be diagonalized simultaneously. Therefore,

        A
      
    
  

{\displaystyle {\mathcal {A}}}

is semi-simple and has a unique basis of primitive idempotents

      J
      
        0
      
    
    ,
    …
    ,
    
      J
      
        n
      
    
  

{\displaystyle J_{0},\ldots ,J_{n}}

. There is another algebra of

    (
    n
    +
    1
    )
    ×
    (
    n
    +
    1
    )
  

{\displaystyle (n+1)\times (n+1)}

matrices which is isomorphic to

        A
      
    
  

{\displaystyle {\mathcal {A}}}

, 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) are the

          (
        
        
          v
          k
        
        
          )
        
      
    
  

{\displaystyle {v \choose 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 yields an association scheme on

    X
    =
    G
  

{\displaystyle X=G}

, with a class Rg for each group element, as follows: for each

    g
    ∈
    G
  

{\displaystyle g\in G}

let

      R
      
        g
      
    
    =
    {
    (
    x
    ,
    y
    )
    ∣
    x
    =
    g
    ∗
    y
    }
  

{\displaystyle R_{g}=\{(x,y)\mid x=g*y\}}

where

    ∗
  

{\displaystyle *}

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 == Block design Bose–Mesner algebra Combinatorial design

== Notes ==