Graphs, designs and codes related to the n-cube

For integers n >= 1, k >= 0, and k <= n, the graph Gamma(k)(n) has vertices the 2(n) vectors of F-2(n) and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular Gamma(1)(n) is the n-cube, usually denoted by Q(n). We examine the binary codes obtained from the adjacency matrices of these graphs when k = 1, 2, 3, following the results obtained for the binary codes of the n-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 Senevirame [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Q, where n 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, 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 Gamma(2)(n) when n equivalent to 0 (mod 4), n >= 8. (C) 2008 Elsevier B.V. All rights reserved.