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.