Graphs, designs and codes related to the nn-cube

For integers n≥1,k≥0n≥1,k≥0, and k≤nk≤n, the graph Γnk has vertices the 2n2n vectors of F2n and adjacency defined by two vectors being adjacent if they differ in kk coordinate positions. In particular Γn1 is the nn-cube, usually denoted by QnQn. We examine the binary codes obtained from the adjacency matrices of these graphs when k=1,2,3k=1,2,3, following the results obtained for the binary codes of the nn-cube in Fish [Washiela Fish, Codes from uniform subset graphs and cyclic products, Ph.D. Thesis, University of the Western Cape, 2007] and Key and Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph QnQn where nn is even, in: T. Shaska, W. C Huffman, D. Joyner, V. Ustimenko (Eds.), Advances in Coding Theory and Cryptology, in: Series on Coding Theory and Cryptology, vol. 2, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007, pp. 152–159]. We find the automorphism groups of the graphs and of their associated neighbourhood designs for k=1,2,3k=1,2,3, and the dimensions of the ternary codes for k=1,2k=1,2. We also obtain 3-PD-sets for the self-dual binary codes from Γn2 when n≡0(mod4), n≥8n≥8.