# American Institute of Mathematical Sciences

May  2011, 5(2): 373-394. doi: 10.3934/amc.2011.5.373

## Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$

 1 Department of Mathematics and Applied Mathematics, University of the Western Cape, 7535 Bellville, South Africa, South Africa, South Africa

Received  May 2010 Revised  June 2010 Published  May 2011

We examine the $p$-ary codes, for any prime $p$, that can be obtained from incidence matrices and line graphs of the Hamming graphs, $H^k(n,m)$, for $k \geq 2$. For $m=2$, we obtain the main parameters of the codes from the incidence matrices, including the minimum weight and the nature of the minimum words. We show that all the codes can be used for full permutation decoding.
Citation: Jennifer D. Key, Washiela Fish, Eric Mwambene. Codes from the incidence matrices and line graphs of Hamming graphs $H^k(n,2)$ for $k \geq 2$. Advances in Mathematics of Communications, 2011, 5 (2) : 373-394. doi: 10.3934/amc.2011.5.373
##### References:
 [1] E. F. Assmus, Jr and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge, 1992.  Google Scholar [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar [3] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, in "Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],'' Springer-Verlag, Berlin, (1989), 495.  Google Scholar [4] J. Cannon, A. Steel and G. White, Linear codes over finite fields, in "Handbook of Magma Functions'' (eds. J. Cannon and W. Bosma), 3951-4023; Magma, Computational Algebra Group, Department of Mathematics, University of Sydney, 2006, V2.13, http://magma.maths.usyd.edu.au/magma Google Scholar [5] W. Fish, "Codes from Uniform Subset Graphs and Cyclic Products,'' Ph.D thesis, University of the Western Cape, 2007. Google Scholar [6] W. Fish, J. D. Key and E. Mwambene, Codes, designs and groups from the Hamming graphs, J. Combin. Inform. System Sci., 34 (2009), 169-182. Google Scholar [7] W. Fish, J. D. Key and E. Mwambene, Graphs, designs and codes related to the $n$-cube, Discrete Math., 309 (2009), 3255-3269. doi: 10.1016/j.disc.2008.09.024.  Google Scholar [8] W. Fish, J. D. Key and E. Mwambene, Binary codes of line graphs from the $n$-cube, J. Symb. Comput., 45 (2010), 800-812. doi: 10.1016/j.jsc.2010.03.012.  Google Scholar [9] W. Fish, J. D. Key and E. Mwambene, Codes from the incidence matrices and line graphs of Hamming graphs, Discrete Math., 310 (2010), 1884-1897. doi: 10.1016/j.disc.2010.02.010.  Google Scholar [10] D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543. doi: 10.1109/TIT.1982.1056504.  Google Scholar [11] W. C. Huffman, Codes and groups, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 1345-1440.  Google Scholar [12] J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682. doi: 10.1016/j.ejc.2004.04.007.  Google Scholar [13] J. D. Key, T. P. McDonough and V. C. Mavron, Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247. doi: 10.1016/j.ffa.2005.05.007.  Google Scholar [14] J. D. Key, J. Moori and B. G. Rodrigues, Codes associated with triangular graphs, and permutation decoding, Int. J. Inform. Coding Theory, 1 (2010), 334-349. doi: 10.1504/IJICOT.2010.032547.  Google Scholar [15] J. D. Key and B. G. Rodrigues, Codes from lattice and related graphs, and permutation decoding, Discrete Appl. Math., 158 (2010), 1807-1815. doi: 10.1016/j.dam.2010.07.003.  Google Scholar [16] J. D. Key and B. G. Rodrigues, Codes from incidence matrices of strongly regular graphs,, in preparation., ().   Google Scholar [17] J. D. Key and P. Seneviratne, Permutation decoding for binary self-dual codes from the graph $Q_n$ where $n$ is even, in "Advances in Coding Theory and Cryptology'' (eds. T. Shaska, W. C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2007), 152-159.  Google Scholar [18] H.-J. Kroll and R. Vincenti, PD-sets related to the codes of some classical varieties, Discrete Math., 301 (2005), 89-105. doi: 10.1016/j.disc.2004.11.020.  Google Scholar [19] J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' Cambridge University Press, Cambridge, 1992.  Google Scholar [20] F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. Google Scholar [21] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1983. Google Scholar [22] R. Peeters., On the $p$-ranks of the adjacency matrices of distance-regular graphs, J. Algebraic Combin., 15 (2002), 127-149. doi: 10.1023/A:1013842904024.  Google Scholar [23] J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  Google Scholar [24] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math., 54 (1932), 154-168. doi: 10.2307/2371086.  Google Scholar

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##### References:
 [1] E. F. Assmus, Jr and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge, 1992.  Google Scholar [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language, J. Symb. Comp., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar [3] A. E. Brouwer, A. M. Cohen and A. Neumaier, Distance-Regular Graphs, in "Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)],'' Springer-Verlag, Berlin, (1989), 495.  Google Scholar [4] J. Cannon, A. Steel and G. White, Linear codes over finite fields, in "Handbook of Magma Functions'' (eds. J. Cannon and W. Bosma), 3951-4023; Magma, Computational Algebra Group, Department of Mathematics, University of Sydney, 2006, V2.13, http://magma.maths.usyd.edu.au/magma Google Scholar [5] W. Fish, "Codes from Uniform Subset Graphs and Cyclic Products,'' Ph.D thesis, University of the Western Cape, 2007. Google Scholar [6] W. Fish, J. D. Key and E. Mwambene, Codes, designs and groups from the Hamming graphs, J. Combin. Inform. System Sci., 34 (2009), 169-182. Google Scholar [7] W. Fish, J. D. Key and E. Mwambene, Graphs, designs and codes related to the $n$-cube, Discrete Math., 309 (2009), 3255-3269. doi: 10.1016/j.disc.2008.09.024.  Google Scholar [8] W. Fish, J. D. Key and E. Mwambene, Binary codes of line graphs from the $n$-cube, J. Symb. Comput., 45 (2010), 800-812. doi: 10.1016/j.jsc.2010.03.012.  Google Scholar [9] W. Fish, J. D. Key and E. Mwambene, Codes from the incidence matrices and line graphs of Hamming graphs, Discrete Math., 310 (2010), 1884-1897. doi: 10.1016/j.disc.2010.02.010.  Google Scholar [10] D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes, IEEE Trans. Inform. Theory, 28 (1982), 541-543. doi: 10.1109/TIT.1982.1056504.  Google Scholar [11] W. C. Huffman, Codes and groups, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 1345-1440.  Google Scholar [12] J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes, European J. Combin., 26 (2005), 665-682. doi: 10.1016/j.ejc.2004.04.007.  Google Scholar [13] J. D. Key, T. P. McDonough and V. C. Mavron, Information sets and partial permutation decoding for codes from finite geometries, Finite Fields Appl., 12 (2006), 232-247. doi: 10.1016/j.ffa.2005.05.007.  Google Scholar [14] J. D. Key, J. Moori and B. G. Rodrigues, Codes associated with triangular graphs, and permutation decoding, Int. J. Inform. Coding Theory, 1 (2010), 334-349. doi: 10.1504/IJICOT.2010.032547.  Google Scholar [15] J. D. Key and B. G. Rodrigues, Codes from lattice and related graphs, and permutation decoding, Discrete Appl. Math., 158 (2010), 1807-1815. doi: 10.1016/j.dam.2010.07.003.  Google Scholar [16] J. D. Key and B. G. Rodrigues, Codes from incidence matrices of strongly regular graphs,, in preparation., ().   Google Scholar [17] J. D. Key and P. Seneviratne, Permutation decoding for binary self-dual codes from the graph $Q_n$ where $n$ is even, in "Advances in Coding Theory and Cryptology'' (eds. T. Shaska, W. C. Huffman, D. Joyner and V. Ustimenko), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, (2007), 152-159.  Google Scholar [18] H.-J. Kroll and R. Vincenti, PD-sets related to the codes of some classical varieties, Discrete Math., 301 (2005), 89-105. doi: 10.1016/j.disc.2004.11.020.  Google Scholar [19] J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' Cambridge University Press, Cambridge, 1992.  Google Scholar [20] F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. Google Scholar [21] F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1983. Google Scholar [22] R. Peeters., On the $p$-ranks of the adjacency matrices of distance-regular graphs, J. Algebraic Combin., 15 (2002), 127-149. doi: 10.1023/A:1013842904024.  Google Scholar [23] J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  Google Scholar [24] H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math., 54 (1932), 154-168. doi: 10.2307/2371086.  Google Scholar
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