Fish, WKey, J DMwambene, E2023-03-032023-03-032009Fish, W., Key, J. D., & Mwambene, E. (2009). Graphs, designs and codes related to the n-cube. Discrete Mathematics, 309(10), 3255-3269. doi:10.1016/j.disc.2008.09.024https//doi.org:/:10.1016/j.disc.2008.09.024http://hdl.handle.net/10566/8519For integers n 1; k 0, and k n, the graph 􀀀 k n has vertices the 2n vectors of Fn 2 and adjacency defined by two vectors being adjacent if they differ in k coordinate positions. In particular 􀀀 1 n is the n-cube, usually denoted by Qn. We examine the binary codes obtained from the adjacency matrices of these graphs when k D 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 Qn 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 D 1; 2; 3, and the dimensions of the ternary codes for k D 1; 2. We also obtain 3-PD-sets for the self-dual binary codes from 􀀀 2 n when n 0 .mod 4/, n 8.enPermutation decodingCodeDesignGraphArticle