Graphs, designs and codes related to the n-cube

For integers n ≥ 1, k ≥ 0, and k ≤ n, the graph Γkn has vertices the 2n vectors of Fn2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Γ1n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following results obtained for the binary codes of the n-cube in Fish [6] and Key and Seneviratne [12]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k = 1, 2, 3, and the dimensions of the ternary codes for k = 1, 2. We also obtain 3-PD-sets for the self-dual binary codes from Γ2n when n ≡ 0 (mod 4), n ≥ 8.