Definition

Recall that the binary relation $R$ on the placed $X$ may be thought of when subset of $X\; \backslash times\; X$.

The k-class Association Scheme occurs as placed of points, X, along using k+1 binary relations $R\_0,R\_1,\backslash ldots,R\_k$ which partition $X\; \backslash times\; X$ and $R\_0\; =\; \backslash$ (we.e. $R\_0$ is the identity relation), such that a as a consequence holds:

There survive $(k+1)^3$ non-negative integers $p\_^l$ with $Zero\; \backslash le\; i,j,l\; \backslash le\; k$ & for any $(x,y)\; \backslash in\; R\_l$ there come exactly $p\_^l$ elements $z\; \backslash in\; X$ such that $(x,z)\; \backslash in\; R\_i$ & $(z,y)\; \backslash in\; R\_j$

Terminology
If $(x,y)\; \backslash in\; R\_i$ you say that $x$ & $y$ come we^th associates. The statistics $p\_^l$ come known as a parameters of the scheme.

Basic Facts
$p\_^0\; =\; I$, we.e. in case $(x,y)\; \backslash in\; R\_0$ so $x\; =\; y$ & a sole $z$ such that $(x,z)\; \backslash in\; R\_0$ is $z=x$ $\backslash sum\_^0\; =\; |X|$, this is because a $R\_i$ partition $X$.

Examples
The Johnson Scheme, denoted J(v,k) is defined when follows. Let S become the placed by using 5 elements. A points of the scheme J(v,k) come a $$ subsets of S using m elements. 2 k-subsets A,B of S come said to become i^th associates while $|A\; \backslash bigcap\; B|\; =\; k\; -\; i\; personally$. The Hamming Scheme, denoted H(n,q) is defined when follows. A points of H(n,q) come a letter q^north vectors of length n across the placed of size letter q. Both north-tuples x,y are said to exist as i^th associates within case a disagree in exactly i co-ordinate. E.g. whenever x = (1,0,1,1), y = (1,1,1,1), z = (0,0,1,1) so x & y come One^st associates, x & z come One^st associates & y & z come Two^nd associates inside H(4,2). A distance-regular graph, G, is an association scheme, by defining two vertices to become i^th associates whenever their few feet away is i.