Codes Related to and Derived from Hamming Graphs
Muthivhi, Thifhelimbilu Ronald
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For integers n, k 2:: 1, and k ~ n, the graph r~has vertices the 2n vectors of lF2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular, r~is the classical n-cube, usually denoted by Hl (n, 2). This study examines the codes (both binary and p-ary for p an odd prime) of the row span of adjacency and incidence matrices of these graphs. We first examine codes of the adjacency matrices of the n-cube. These have been considered in . We then consider codes generated by both incidence and adjacency matrices of the Hamming graphs Hl(n,3) . We will also consider codes of the line graphs of the n-cube as in . Further, the automorphism groups of the codes, designs and graphs will be examined, highlighting where there is an interplay. Where possible, suitable permutation decoding sets will be given.