Generalized hypercube graph Qn(S), graph products and self-orthogonal codes

A generalized hypercube graph Qn(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 products G1 = Q_n(S) ˄ Q_1;G3 = Q_n(S) ^ Q_1, G_3 = Q_n(S)[Q1] and 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.