Graphs, designs and codes related to the [formula omitted]-cube

Abstract

For integers n ≥ 1 , k ≥ 0 , and k ≤ n , the graph Γ n k 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 Γ n 1 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 Seneviratne [J.D. Key, P. Seneviratne, Permutation decoding for binary self-dual codes from the graph Q n 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 Γ n 2 when n ≡ 0 ( mod 4 ) , n ≥ 8 .