Association scheme

<h2 id="definition">Definition</h2>
An n-class association scheme consists of a <a href="/facts/Set_(mathematics)/BfucfHMq">set</a> X together with a <a href="/facts/Partition_of_a_set/VeP9g8CA">partition</a> S of X × X into n + 1 <a href="/facts/Binary_relation/r9BwGgaK">binary relations</a>, R0, R1, ..., Rn which satisfy:

<ul><li>
 
 
 
 
 R
 
 0
 
 
 =
 {
 (
 x
 ,
 x
 )
 :
 x
 ∈
 X
 }
 
 
 {\displaystyle R_{0}=\{(x,x):x\in X\}}
 
; it is called the <a href="/facts/Identity_relation/wh0QoWzS">identity relation</a>.</li>
<li>Defining 
 
 
 
 
 R
 
 ∗
 
 
 :=
 {
 (
 x
 ,
 y
 )
 :
 (
 y
 ,
 x
 )
 ∈
 R
 }
 
 
 {\displaystyle R^{*}:=\{(x,y):(y,x)\in R\}}
 
, if R in S, then R* in S.</li>
<li>If 
 
 
 
 (
 x
 ,
 y
 )
 ∈
 
 R
 
 k
 
 
 
 
 {\displaystyle (x,y)\in R_{k}}
 
, the number of 
 
 
 
 z
 ∈
 X
 
 
 {\displaystyle z\in X}
 
 such that 
 
 
 
 (
 x
 ,
 z
 )
 ∈
 
 R
 
 i
 
 
 
 
 {\displaystyle (x,z)\in R_{i}}
 
 and 
 
 
 
 (
 z
 ,
 y
 )
 ∈
 
 R
 
 j
 
 
 
 
 {\displaystyle (z,y)\in R_{j}}
 
 is a constant 
 
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 
 
 {\displaystyle p_{ij}^{k}}
 
 depending on 
 
 
 
 i
 
 
 {\displaystyle i}
 
, 
 
 
 
 j
 
 
 {\displaystyle j}
 
, 
 
 
 
 k
 
 
 {\displaystyle k}
 
 but not on the particular choice of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
.</li></ul>
An association scheme is commutative if 
 
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 =
 
 p
 
 j
 i
 
 
 k
 
 
 
 
 {\displaystyle p_{ij}^{k}=p_{ji}^{k}}
 
 for all 
 
 
 
 i
 
 
 {\displaystyle i}
 
, 
 
 
 
 j
 
 
 {\displaystyle j}
 
 and 
 
 
 
 k
 
 
 {\displaystyle 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 <a href="/facts/Commutative_group/FfWwmx4R">commutative group</a>.
A symmetric association scheme is one in which each 
 
 
 
 
 R
 
 i
 
 
 
 
 {\displaystyle R_{i}}
 
 is a <a href="/facts/Symmetric_relation/ekCWDIAH">symmetric relation</a>. That is:

<ul><li>if (x, y) ∈ Ri, then (y, x) ∈ Ri. (Or equivalently, R* = R.)</li></ul>
Every symmetric association scheme is commutative.
Two points x and y are called i th associates if 
 
 
 
 (
 x
 ,
 y
 )
 ∈
 
 R
 
 i
 
 
 
 
 {\displaystyle (x,y)\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
 
 
 {\displaystyle 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
 
 
 {\displaystyle z}
 
 which are both i th associates of 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and j th associates of 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is a constant 
 
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 
 
 {\displaystyle p_{ij}^{k}}
 
.

<h3>Graph interpretation and adjacency matrices</h3>
A symmetric association scheme can be visualized as a <a href="/facts/Complete_graph/AYD6hUqI">complete graph</a> with labeled edges. The graph has 
 
 
 
 v
 
 
 {\displaystyle v}
 
 vertices, one for each point of 
 
 
 
 X
 
 
 {\displaystyle X}
 
, and the edge joining vertices 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 is labeled 
 
 
 
 i
 
 
 {\displaystyle i}
 
 if 
 
 
 
 x
 
 
 {\displaystyle x}
 
 and 
 
 
 
 y
 
 
 {\displaystyle y}
 
 are 
 
 
 
 i
 
 
 {\displaystyle i}
 
 th associates. Each edge has a unique label, and the number of triangles with a fixed base labeled 
 
 
 
 k
 
 
 {\displaystyle k}
 
 having the other edges labeled 
 
 
 
 i
 
 
 {\displaystyle i}
 
 and 
 
 
 
 j
 
 
 {\displaystyle j}
 
 is a constant 
 
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 
 
 {\displaystyle p_{ij}^{k}}
 
, depending on 
 
 
 
 i
 ,
 j
 ,
 k
 
 
 {\displaystyle i,j,k}
 
 but not on the choice of the base. In particular, each vertex is incident with exactly 
 
 
 
 
 p
 
 i
 i
 
 
 0
 
 
 =
 
 v
 
 i
 
 
 
 
 {\displaystyle p_{ii}^{0}=v_{i}}
 
 edges labeled 
 
 
 
 i
 
 
 {\displaystyle i}
 
; 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
 is the <a href="/facts/Adjacency_relation/kw3eIBUe">valency</a> of the <a href="/facts/Relation_(mathematics)/HjblDaMy">relation</a> 
 
 
 
 
 R
 
 i
 
 
 
 
 {\displaystyle R_{i}}
 
. There are also loops labeled 
 
 
 
 0
 
 
 {\displaystyle 0}
 
 at each vertex 
 
 
 
 x
 
 
 {\displaystyle x}
 
, corresponding to 
 
 
 
 
 R
 
 0
 
 
 
 
 {\displaystyle R_{0}}
 
.
The <a href="/facts/Relation_(mathematics)/HjblDaMy">relations</a> are described by their <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrices</a>. 
 
 
 
 
 A
 
 i
 
 
 
 
 {\displaystyle A_{i}}
 
 is the adjacency matrix of 
 
 
 
 
 R
 
 i
 
 
 
 
 {\displaystyle R_{i}}
 
 for 
 
 
 
 i
 =
 0
 ,
 …
 ,
 n
 
 
 {\displaystyle i=0,\ldots ,n}
 
 and is a v × v <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> with rows and columns labeled by the points of 
 
 
 
 X
 
 
 {\displaystyle X}
 
.

(
            
              A
              
                i
              
            
            )
          
          
            x
            ,
            y
          
        
        =
        
          
            {
            
              
                
                  1
                  ,
                
                
                  
                    
                      if 
                    
                  
                  (
                  x
                  ,
                  y
                  )
                  ∈
                  
                    R
                    
                      i
                    
                  
                  ,
                
              
              
                
                  0
                  ,
                
                
                  
                    
                      otherwise.
                    
                  
                
              
            
            
          
        
        
        (
        1
        )
      
    
    {\displaystyle \left(A_{i}\right)_{x,y}={\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
 
 
 
 
 {\displaystyle A_{i}}
 
 are v × v <a href="/facts/(0%2c1)-matrix/pS4BnBzj">(0,1)-matrices</a> which satisfy

I. 
 
 
 
 
 A
 
 i
 
 
 
 
 {\displaystyle A_{i}}
 
 is symmetric,
II. 
 
 
 
 
 ∑
 
 i
 =
 0
 
 
 n
 
 
 
 A
 
 i
 
 
 =
 J
 
 
 {\displaystyle \sum _{i=0}^{n}A_{i}=J}
 
 (the all-ones matrix),
III. 
 
 
 
 
 A
 
 0
 
 
 =
 I
 
 
 {\displaystyle A_{0}=I}
 
,
IV. 
 
 
 
 
 A
 
 i
 
 
 
 A
 
 j
 
 
 =
 
 ∑
 
 k
 =
 0
 
 
 n
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 
 A
 
 k
 
 
 =
 
 A
 
 j
 
 
 
 A
 
 i
 
 
 ,
 i
 ,
 j
 =
 0
 ,
 …
 ,
 n
 
 
 {\displaystyle A_{i}A_{j}=\sum _{k=0}^{n}p_{ij}^{k}A_{k}=A_{j}A_{i},i,j=0,\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
 
 
 
 
 {\displaystyle A_{i}}
 
 contain 
 
 
 
 
 v
 
 i
 
 
 
 
 {\displaystyle v_{i}}
 
 
 
 
 
 1
 
 
 {\displaystyle 1}
 
's:

A
          
            i
          
        
        J
        =
        J
        
          A
          
            i
          
        
        =
        
          v
          
            i
          
        
        J
        .
        
        (
        2
        )
      
    
    {\displaystyle A_{i}J=JA_{i}=v_{i}J.\qquad (2)}

<h3>Terminology</h3>
<ul><li>The numbers 
 
 
 
 
 p
 
 i
 j
 
 
 k
 
 
 
 
 {\displaystyle p_{ij}^{k}}
 
 are called the parameters of the scheme. They are also referred to as the structural constants.</li></ul>
<h2 id="history">History</h2>
The term association scheme is due to (Bose & Shimamoto 1952) but the concept is already inherent in (Bose & Nair 1939).<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> 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 <a href="/facts/Bose%25E2%2580%2593Mesner_algebra/1ovWTsIG">Bose–Mesner algebra</a>. 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 class="footnote-ref" id="fnref:10" href="#fn:10">10</a> 
A generalization called <a href="/facts/Coherent_configuration/MLqL2NEB">coherent configurations</a> has been studied by D. G. Higman.

<h2 id="basic-facts">Basic facts</h2>
<ul><li>
 
 
 
 
 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}
 
.</li>
<li>
 
 
 
 
 ∑
 
 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}
 
.</li></ul>
<h2 id="the-bosemesner-algebra">The Bose–Mesner algebra</h2>
The <a href="/facts/Adjacency_matrix/jYXNhYGs">adjacency matrices</a> 
 
 
 
 
 A
 
 i
 
 
 
 
 {\displaystyle A_{i}}
 
 of the <a href="/facts/Graph_(discrete_mathematics)/kw3eIBUe">graphs</a> 
 
 
 
 
 (
 
 X
 ,
 
 R
 
 i
 
 
 
 )
 
 
 
 {\displaystyle \left(X,R_{i}\right)}
 
 generate a <a href="/facts/Commutative_algebra_(structure)/DRfoF6KV">commutative</a> and <a href="/facts/Associative_algebra/DRfoF6KV">associative algebra</a> 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 (over the <a href="/facts/Real_number/R02gw5Pb">real</a> or <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a>) both for the <a href="/facts/Matrix_product/lYDB6Ro2">matrix product</a> and the <a href="/facts/Hadamard_product_(matrices)/NjAUd4o4">pointwise product</a>. This associative, commutative algebra is called the <a href="/facts/Bose%25E2%2580%2593Mesner_algebra/1ovWTsIG">Bose–Mesner algebra</a> of the association scheme.
Since the matrices in 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 are <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a> and <a href="/facts/Commuting_matrices/R1wGesmf">commute</a> with each other, they can be <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonalized</a> simultaneously. Therefore, 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
 is <a href="/facts/Semisimple_operator/J3Jdl8Qh">semi-simple</a> and has a unique basis of primitive <a href="/facts/Idempotent/khbSpexv">idempotents</a> 
 
 
 
 
 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 <a href="/facts/Isomorphic/pj8KgkxU">isomorphic</a> to 
 
 
 
 
 
 A
 
 
 
 
 {\displaystyle {\mathcal {A}}}
 
, and is often easier to work with.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Johnson_scheme/KbpjeLzS">Johnson scheme</a>, 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}}
 
 <a href="/facts/Subset/SJGERmbA">subsets</a> of S with k elements. Two k-element subsets A, B of S are i th associates when their intersection has size k − i.</li>
<li>The <a href="/facts/Hamming_scheme/xxs8GjWs">Hamming scheme</a>, denoted by H(n, q), is defined as follows. The points of H(n, q) are the qn ordered n-<a href="/facts/Tuple/BI1HIxL8">tuples</a> 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).</li>
<li>A <a href="/facts/Distance-regular_graph/pGAuLWYL">distance-regular graph</a>, G, forms an association scheme by defining two vertices to be i th associates if their distance is i.</li>
<li>A <a href="/facts/Finite_group/vv2tvogB">finite group</a> 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 <a href="/facts/Operation_(mathematics)/Yplv602Q">operation</a>. The class of the <a href="/facts/Identity_element/9nss3xHY">group identity</a> is R0. This association scheme is commutative <a href="/facts/If_and_only_if/bYSxGJ66">if and only if</a> G is <a href="/facts/Abelian_group/FfWwmx4R">abelian</a>.</li>
<li>A specific 3-class association scheme:<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a></li></ul>
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.
<table><tbody><tr><th> </th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th></tr><tr><td>1</td><td>  0 </td><td>  1  </td><td>  1  </td><td>  2  </td><td>  3 </td><td>  3 </td></tr><tr><td>2</td><td>  1  </td><td>  0  </td><td>  1  </td><td>  3 </td><td>  2 </td><td>  3 </td></tr><tr><td>3</td><td>  1  </td><td>  1  </td><td>  0 </td><td>  3 </td><td>  3 </td><td>  2 </td></tr><tr><td>4</td><td>  2 </td><td>  3 </td><td>  3 </td><td>  0 </td><td>  1  </td><td>  1  </td></tr><tr><td>5</td><td>  3 </td><td>  2 </td><td>  3 </td><td>  1  </td><td> 0 </td><td>  1  </td></tr><tr><td>6</td><td>  3  </td><td>  3 </td><td>  2 </td><td>  1  </td><td>  1  </td><td> 0 </td></tr></tbody></table>
<h2 id="coding-theory">Coding theory</h2>
The <a href="/facts/Hamming_scheme/xxs8GjWs">Hamming scheme</a> and the <a href="/facts/Johnson_scheme/KbpjeLzS">Johnson scheme</a> are of major significance in classical <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>.
In <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>, association scheme theory is mainly concerned with the <a href="/facts/Hamming_distance/MgdtbYPo">distance</a> of a <a href="/facts/Code/CSntvnEo">code</a>. The <a href="/facts/Linear_programming/GduXFQxT">linear programming</a> method produces upper bounds for the size of a <a href="/facts/Code/CSntvnEo">code</a> with given minimum <a href="/facts/Hamming_distance/MgdtbYPo">distance</a>, and lower bounds for the size of a <a href="/facts/T-design/R523MRYh">design</a> with a given strength. The most specific results are obtained in the case where the underlying association scheme satisfies certain <a href="/facts/Polynomial/Lzak8VVx">polynomial</a> properties; this leads one into the realm of <a href="/facts/Orthogonal_polynomials/9cyRbDYU">orthogonal polynomials</a>. In particular, some universal bounds are derived for <a href="/facts/Code/CSntvnEo">codes</a> and <a href="/facts/T-design/R523MRYh">designs</a> in polynomial-type association schemes.
In classical <a href="/facts/Coding_theory/SBqStoGL">coding theory</a>, dealing with <a href="/facts/Code/CSntvnEo">codes</a> in a <a href="/facts/Hamming_scheme/xxs8GjWs">Hamming scheme</a>, the MacWilliams transform involves a family of orthogonal polynomials known as the <a href="/facts/Krawtchouk_polynomials/Uh7LlP0h">Krawtchouk polynomials</a>. These polynomials give the <a href="/facts/Eigenvalue/8TjEoT8u">eigenvalues</a> of the distance relation matrices of the <a href="/facts/Hamming_scheme/xxs8GjWs">Hamming scheme</a>.

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Block_design/R523MRYh">Block design</a></li>
<li><a href="/facts/Bose%25E2%2580%2593Mesner_algebra/1ovWTsIG">Bose–Mesner algebra</a></li>
<li><a href="/facts/Combinatorial_design/V6dv0Jop">Combinatorial design</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Rosemary_A._Bailey/TeDyH54y">Bailey, Rosemary A.</a> (2004), <a href="http://www.maths.qmul.ac.uk/~rab/Asbook">Association Schemes: Designed Experiments, Algebra and Combinatorics</a>, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-82446-0, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2047311">2047311</a>. (Chapters from preliminary draft are <a href="http://www.maths.qmw.ac.uk/~rab">available on-line</a>.)</li>
<li>Bannai, Eiichi; Ito, Tatsuro (1984), Algebraic combinatorics I: Association schemes, Menlo Park, CA: Benjamin/Cummings, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8053-0490-8, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882540">0882540</a></li>
<li><a href="/facts/R._C._Bose/RI0WHXIV">Bose, R. C.</a>; Mesner, D. M. (1959), <a href="http://projecteuclid.org/euclid.aoms/1177706356">"On linear associative algebras corresponding to association schemes of partially balanced designs"</a>, <a href="/facts/Annals_of_Mathematical_Statistics/R8H9bM4d">Annals of Mathematical Statistics</a>, 30 (1): 21–38, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1214%2Faoms%2F1177706356">10.1214/aoms/1177706356</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2237117">2237117</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0102157">0102157</a></li>
<li><a href="/facts/R._C._Bose/RI0WHXIV">Bose, R. C.</a>; Nair, K. R. (1939), "Partially balanced incomplete block designs", <a href="/facts/Sankhya_(journal)/8lxTxfts">Sankhyā</a>, 4 (3): 337–372, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/40383923">40383923</a></li>
<li><a href="/facts/R._C._Bose/RI0WHXIV">Bose, R. C.</a>; Shimamoto, T. (1952), "Classification and analysis of partially balanced incomplete block designs with two associate classes", Journal of the American Statistical Association, 47 (258): 151–184, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1080%2F01621459.1952.10501161">10.1080/01621459.1952.10501161</a></li>
<li>Camion, P. (1998), "18. Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding", in Pless, V.S.; Huffman, W.C.; Brualdi, R.A. (eds.), Handbook of Coding Theory, vol. 1, Elsevier, pp. 1441–, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-444-50088-5</li>
<li>Delsarte, P. (1973), "An Algebraic Approach to the Association Schemes of Coding Theory", Philips Research Reports (Supplement No. 10), <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/641852316">641852316</a></li>
<li>Delsarte, P.; Levenshtein, V. I. (1998). "Association schemes and coding theory". IEEE Transactions on Information Theory. 44 (6): 2477–2504. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F18.720545">10.1109/18.720545</a>.</li>
<li>Dembowski, P. (1968), <a href="https://books.google.com/books?id=NJy_iTT_wGMC">Finite Geometries</a>, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-540-61786-0</li>
<li><a href="/facts/Chris_Godsil/ygA2pwGf">Godsil, C. D.</a> (1993), Algebraic Combinatorics, New York: Chapman and Hall, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-04131-6, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1220704">1220704</a></li>
<li>MacWilliams, F.J.; Sloane, N.J.A. (1977), <a href="https://books.google.com/books?id=nv6WCJgcjxcC&pg=PP1">The Theory of Error Correcting Codes</a>, North-Holland Mathematical Library, vol. 16, Elsevier, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-444-85010-2</li>
<li><a href="/facts/Anne_Penfold_Street/JHUjj7AU">Street, Anne Penfold</a>; <a href="/facts/Deborah_Street/6DbqKEnX">Street, Deborah J.</a> (1987), <a href="/facts/Combinatorics_of_Experimental_Design/Rqod0clC">Combinatorics of Experimental Design</a>, Oxford U. P. [Clarendon], <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853256-3</li></ul>
<ul><li>van Lint, J.H.; Wilson, R.M. (1992), A Course in Combinatorics, Cambridge University Press, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-00601-5</li>
<li>Zieschang, Paul-Hermann (2005a), <a href="https://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf">"Association Schemes: Designed Experiments, Algebra and Combinatorics by Rosemary A. Bailey, Review"</a> (PDF), Bulletin of the American Mathematical Society, 43 (2): 249–253, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2FS0273-0979-05-01077-3">10.1090/S0273-0979-05-01077-3</a></li>
<li>Zieschang, Paul-Hermann (2005b), Theory of association schemes, Springer, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-26136-2</li>
<li>Zieschang, Paul-Hermann (2006), "The exchange condition for association schemes", <a href="/facts/Israel_Journal_of_Mathematics/mSGtfzpS">Israel Journal of Mathematics</a>, 151 (3): 357–380, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF02777367">10.1007/BF02777367</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2214129">2214129</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:120009352">120009352</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Bailey 2004, pg. 387 - Bailey, Rosemary A. (2004), Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, ISBN 978-0-521-82446-0, MR 2047311 <a href="http://www.maths.qmul.ac.uk/~rab/Asbook" target="_blank">http://www.maths.qmul.ac.uk/~rab/Asbook</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Bose & Mesner 1959 - Bose, R. C.; Mesner, D. M. (1959), "On linear associative algebras corresponding to association schemes of partially balanced designs", Annals of Mathematical Statistics, 30 (1): 21–38, doi:10.1214/aoms/1177706356, JSTOR 2237117, MR 0102157 <a href="http://projecteuclid.org/euclid.aoms/1177706356" target="_blank">http://projecteuclid.org/euclid.aoms/1177706356</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Bose & Nair 1939 - Bose, R. C.; Nair, K. R. (1939), "Partially balanced incomplete block designs", Sankhyā, 4 (3): 337–372, JSTOR 40383923 <a href="https://www.jstor.org/stable/40383923" target="_blank">https://www.jstor.org/stable/40383923</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Bannai & Ito 1984 - Bannai, Eiichi; Ito, Tatsuro (1984), Algebraic combinatorics I: Association schemes, Menlo Park, CA: Benjamin/Cummings, ISBN 0-8053-0490-8, MR 0882540 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882540" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0882540</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Godsil 1993 - Godsil, C. D. (1993), Algebraic Combinatorics, New York: Chapman and Hall, ISBN 0-412-04131-6, MR 1220704 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1220704" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1220704</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Bailey 2004, pg. 387 - Bailey, Rosemary A. (2004), Association Schemes: Designed Experiments, Algebra and Combinatorics, Cambridge University Press, ISBN 978-0-521-82446-0, MR 2047311 <a href="http://www.maths.qmul.ac.uk/~rab/Asbook" target="_blank">http://www.maths.qmul.ac.uk/~rab/Asbook</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Zieschang 2005b - Zieschang, Paul-Hermann (2005b), Theory of association schemes, Springer, ISBN 3-540-26136-2 <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Zieschang 2005a - Zieschang, Paul-Hermann (2005a), "Association Schemes: Designed Experiments, Algebra and Combinatorics by Rosemary A. Bailey, Review" (PDF), Bulletin of the American Mathematical Society, 43 (2): 249–253, doi:10.1090/S0273-0979-05-01077-3 <a href="https://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf" target="_blank">https://www.ams.org/bull/2006-43-02/S0273-0979-05-01077-3/S0273-0979-05-01077-3.pdf</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Dembowski 1968, pg. 281, footnote 1 - Dembowski, P. (1968), Finite Geometries, Springer, ISBN 978-3-540-61786-0 <a href="https://books.google.com/books?id=NJy_iTT_wGMC" target="_blank">https://books.google.com/books?id=NJy_iTT_wGMC</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Bannai & Ito 1984, pg. vii - Bannai, Eiichi; Ito, Tatsuro (1984), Algebraic combinatorics I: Association schemes, Menlo Park, CA: Benjamin/Cummings, ISBN 0-8053-0490-8, MR 0882540 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0882540" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0882540</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Street & Street 1987, pg. 238 - Street, Anne Penfold; Street, Deborah J. (1987), Combinatorics of Experimental Design, Oxford U. P. [Clarendon], ISBN 0-19-853256-3 <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
</ol>

Association scheme open-in-new

Association scheme