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The Hamming scheme, named after Richard Hamming, is also known as the hyper-cubic association scheme, and it is the most important example for coding theory. In this scheme $X={\mathcal{ℱ}}^n$, the set of binary vectors of length $n$, and two vectors $x$, $y\in {\mathcal{ℱ}}^n$ are $i$-th associates if they have Hamming distance $i$ apart.

Recall that an association scheme is 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 $c_{ijk}$, depending on $i,j,k$ but not on the choice of the base. In particular, each vertex is incident with exactly $c_{ii0}=v_i$ edges labeled $i$; $v_i$ is the valency of the relation $R_i$. The $c_{ijk}$ in a Hamming scheme are given by

$c_{ijk}=\{\begin{array}{ll}(\genfrac{}{}{0ex}{}{k}{\frac{i-j+k}{2}})(\genfrac{}{}{0ex}{}{n-k}{\frac{i+j-k}{2}}),& \text{if }i+j-k\text{ is even,}\\ 0,& \text{if }i+j-k\text{ is odd.}\end{array}$

Here, $v=\left|X\right|=2^n$ and $v_i=(\genfrac{}{}{0ex}{}{n}{i})$. The matrices in the Bose-Mesner algebra are $2^n\times 2^n$ matrices, with rows and columns labeled by vectors $x\in {\mathcal{ℱ}}^n$. In particular the $(x,y)$-th entry of $D_k$ is $1$ if and only if $d_H(x,y)=k$.

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