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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 |
References:
[1] |
E. F. Assmus, Jr and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge, 1992. |
[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. |
[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. |
[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 |
[5] |
W. Fish, "Codes from Uniform Subset Graphs and Cyclic Products,'' Ph.D thesis, University of the Western Cape, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. D. Key and B. G. Rodrigues, Codes from incidence matrices of strongly regular graphs, in preparation. |
[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. |
[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. |
[19] |
J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' Cambridge University Press, Cambridge, 1992. |
[20] |
F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. |
[21] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1983. |
[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. |
[23] |
J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411. |
[24] |
H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math., 54 (1932), 154-168.
doi: 10.2307/2371086. |
show all references
References:
[1] |
E. F. Assmus, Jr and J. D. Key, "Designs and Their Codes,'' Cambridge University Press, Cambridge, 1992. |
[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. |
[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. |
[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 |
[5] |
W. Fish, "Codes from Uniform Subset Graphs and Cyclic Products,'' Ph.D thesis, University of the Western Cape, 2007. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[16] |
J. D. Key and B. G. Rodrigues, Codes from incidence matrices of strongly regular graphs, in preparation. |
[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. |
[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. |
[19] |
J. H. van Lint and R. M. Wilson, "A Course in Combinatorics,'' Cambridge University Press, Cambridge, 1992. |
[20] |
F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964), 485-505. |
[21] |
F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North-Holland, Amsterdam, 1983. |
[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. |
[23] |
J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411. |
[24] |
H. Whitney, Congruent graphs and the connectivity of graphs, Amer. J. Math., 54 (1932), 154-168.
doi: 10.2307/2371086. |
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