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

A generalized hypercube graph $\Q_n(S)$ has $\F_{2}^{n}=\{0,1\}^n$ as the vertex set and two vertices being adjacent whenever their mutual Hamming distance belongs to $S$, where $n \ge 1$ and $S\subseteq \{1,2,\ldots, n\}$. The graph $\Q_n(\{1\})$ is the $n$-cube, usually denoted by $\Q_n$. We study graph boolean products $G_1 = \Q_n(S)\times \Q_1, G_2 = \Q_{n}(S)\wedge \Q_1$, $G_3 = \Q_{n}(S)[\Q_1]$ and show that binary codes from neighborhood designs of $G_1, G_2$ and $G_3$ are self-orthogonal for all choices of $n$ and $S$. More over, we show that the class of codes $C_1$ 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.