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. 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


 * $$c_{ijk} = \begin{cases} \dbinom{k}{\frac{1}{2}(i-j+k)} \dbinom{n-k}{\frac{1}{2}(i-j+k)} & i+j-k \equiv 0 \pmod 2 \\ \\ 0& i+j-k \equiv 1 \pmod 2 \end{cases}$$

Here, $$v=|X|=2^n$$ and $$v_i=\tbinom{n}{i}.$$ 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 $$d_{H}(x,y)=k.$$