Generalized hypercube graph $\Q_n(S)$, graph products and self-orthogonal codes

Abstract

A generalized hypercube graph Q n (S) has F n 2 = {0, 1} n as the vertex set and two vertices being adjacent whenever their mutual Hamming distance belongs to S, where n ≥ 1 and S ⊆ {1, 2, . . . , n}. The graph Q n ({1}) is the n-cube, usually denoted by Q n . We study graph boolean productsand show that binary codes from neighborhood designs of G1, G2 and G3 are self-orthogonal for all choices of n and S. More over, we show that the class of codes C1 are self-dual. Further we find subgroups of the automorphism group of these graphs and use these subgroups to obtain PD-sets for permutation decoding. As an example we find a full error-correcting PD set for the binary [32, 16,8] extremal self-dual code.