Hamming scheme

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.^[1][2][3] 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 are 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

$c_{ijk} = \begin{cases} \dbinom{k}{\frac{i-j+k}{2}}\dbinom{n-k}{\frac{i+j-k}{2}}, & \text{if } i+j-k \text{ is even,} \\ \;\;\;\;\;\;\;\;\;\;0\;\;\;\;\;\;\;\;\;\;\;,\;\; & \text{if } i+j-k \text{ is odd.} \end{cases}$

Here, $v=\left|X\right|=2^{n}$ and $v_{i}=\binom{n}{i}$. 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 $\left(x,y\right)$-th entry of $D_{k}$ is $1$ if and only if $d_{H}(x,y)=k$.

References

1. P. Delsarte and V. I. Levenshtein, “Association schemes and coding theory,“ IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2477–2504, 1998.
2. P. Camion, "Codes and Association Schemes: Basic Properties of Association Schemes Relevant to Coding," in Handbook of Coding Theory, V. S. Pless and W. C. Huffman, Eds., Elsevier, The Netherlands, 1998.
3. F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier, New York, 1978.