Hamming Scheme

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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, yin 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 cijk, depending on i,j,k but not on the choice of the base. In particular, each vertex is incident with exactly cii0 = vi edges labeled i; vi is the valency of the relation Ri.