# American Institute of Mathematical Sciences

May  2015, 9(2): 211-232. doi: 10.3934/amc.2015.9.211

## Binary codes from reflexive uniform subset graphs on $3$-sets

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

Received  March 2014 Published  May 2015

We examine the binary codes $C_2(A_i+I)$ from matrices $A_i+I$ where $A_i$ is an adjacency matrix of a uniform subset graph $\Gamma(n,3,i)$ of $3$-subsets of a set of size $n$ with adjacency defined by subsets meeting in $i$ elements of $\Omega$, where $0 \le i \le 2$. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the $C_2(A_i+I)$ are also examined. We obtain partial PD-sets for some of the codes, for permutation decoding.
Citation: Washiela Fish, Jennifer D. Key, Eric Mwambene. Binary codes from reflexive uniform subset graphs on $3$-sets. Advances in Mathematics of Communications, 2015, 9 (2) : 211-232. doi: 10.3934/amc.2015.9.211
##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and their Codes,, Cambridge University Press, (1992). Google Scholar [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language,, J. Symb. Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar [3] J. Cannon, A. Steel and G. White, Linear codes over finite fields,, in Handbook of Magma Functions (eds. J. Cannon and W. Bosma), (2006), 3951. Google Scholar [4] L. Chikamai, Linear Codes Obtained from $2$-Modular Representations of Some Finite Simple Groups,, Ph.D thesis, (2013). Google Scholar [5] P. Dankelmann, J. D. Key and B. G. Rodrigues, A characterization of graphs by codes from their incidence matrices,, Electron. J. Combin., 20 (2013). Google Scholar [6] P. Dankelmann, J. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs,, Des. Codes Crypt., 68 (2013), 373. doi: 10.1007/s10623-011-9594-x. Google Scholar [7] W. Fish, Codes from Uniform Subset Graphs and Cycle Products,, Ph.D thesis, (2007). Google Scholar [8] W. Fish, J. D. Key and E. Mwambene, Ternary codes from reflexive graphs on $3$-sets,, Appl. Algebra Engrg. Comm. Comput., 25 (2014), 363. doi: 10.1007/s00200-014-0233-4. Google Scholar [9] W. Fish, J. D. Key and E. Mwambene, Self-orthogonal binary codes from odd graphs,, Util. Math., (). Google Scholar [10] D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes,, IEEE Trans. Inf. Theory, 28 (1982), 541. doi: 10.1109/TIT.1982.1056504. Google Scholar [11] W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs,, Des. Codes Crypt., 17 (1999), 187. doi: 10.1023/A:1008353723204. Google Scholar [12] W. C. Huffman, Codes and groups,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 1345. Google Scholar [13] J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes,, European J. Combin., 26 (2005), 665. doi: 10.1016/j.ejc.2004.04.007. Google Scholar [14] J. D. Key, J. Moori and B. G. Rodrigues, Binary codes from graphs on triples,, Discrete Math., 282 (2004), 171. doi: 10.1016/j.disc.2003.12.004. Google Scholar [15] J. D. Key, J. Moori and B. G. Rodrigues, Partial permutation decoding of some binary codes from graphs on triples,, Ars Combin., 79 (2006), 11. Google Scholar [16] J. D. Key, J. Moori and B. G. Rodrigues, Ternary codes from graphs on triples,, Discrete Math., 309 (2009), 4663. doi: 10.1016/j.disc.2008.05.032. Google Scholar [17] H.-J. Kroll and R. Vincenti, PD-sets related to the codes of some classical varieties,, Discrete Math., 301 (2005), 89. doi: 10.1016/j.disc.2004.11.020. Google Scholar [18] F. J. MacWilliams, Permutation decoding of systematic codes,, Bell System Tech. J., 43 (1964), 485. Google Scholar [19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1983). Google Scholar [20] R. Peeters, On the $p$-ranks of the adjacency matrices of distance-regular graphs,, J. Algebr. Combin., 15 (2002), 127. doi: 10.1023/A:1013842904024. Google Scholar [21] J. Schönheim, On coverings,, Pacific J. Math., 14 (1964), 1405. Google Scholar

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##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and their Codes,, Cambridge University Press, (1992). Google Scholar [2] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I: The user language,, J. Symb. Comput., 24 (1997), 235. doi: 10.1006/jsco.1996.0125. Google Scholar [3] J. Cannon, A. Steel and G. White, Linear codes over finite fields,, in Handbook of Magma Functions (eds. J. Cannon and W. Bosma), (2006), 3951. Google Scholar [4] L. Chikamai, Linear Codes Obtained from $2$-Modular Representations of Some Finite Simple Groups,, Ph.D thesis, (2013). Google Scholar [5] P. Dankelmann, J. D. Key and B. G. Rodrigues, A characterization of graphs by codes from their incidence matrices,, Electron. J. Combin., 20 (2013). Google Scholar [6] P. Dankelmann, J. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs,, Des. Codes Crypt., 68 (2013), 373. doi: 10.1007/s10623-011-9594-x. Google Scholar [7] W. Fish, Codes from Uniform Subset Graphs and Cycle Products,, Ph.D thesis, (2007). Google Scholar [8] W. Fish, J. D. Key and E. Mwambene, Ternary codes from reflexive graphs on $3$-sets,, Appl. Algebra Engrg. Comm. Comput., 25 (2014), 363. doi: 10.1007/s00200-014-0233-4. Google Scholar [9] W. Fish, J. D. Key and E. Mwambene, Self-orthogonal binary codes from odd graphs,, Util. Math., (). Google Scholar [10] D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes,, IEEE Trans. Inf. Theory, 28 (1982), 541. doi: 10.1109/TIT.1982.1056504. Google Scholar [11] W. H. Haemers, R. Peeters and J. M. van Rijckevorsel, Binary codes of strongly regular graphs,, Des. Codes Crypt., 17 (1999), 187. doi: 10.1023/A:1008353723204. Google Scholar [12] W. C. Huffman, Codes and groups,, in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), (1998), 1345. Google Scholar [13] J. D. Key, T. P. McDonough and V. C. Mavron, Partial permutation decoding for codes from finite planes,, European J. Combin., 26 (2005), 665. doi: 10.1016/j.ejc.2004.04.007. Google Scholar [14] J. D. Key, J. Moori and B. G. Rodrigues, Binary codes from graphs on triples,, Discrete Math., 282 (2004), 171. doi: 10.1016/j.disc.2003.12.004. Google Scholar [15] J. D. Key, J. Moori and B. G. Rodrigues, Partial permutation decoding of some binary codes from graphs on triples,, Ars Combin., 79 (2006), 11. Google Scholar [16] J. D. Key, J. Moori and B. G. Rodrigues, Ternary codes from graphs on triples,, Discrete Math., 309 (2009), 4663. doi: 10.1016/j.disc.2008.05.032. Google Scholar [17] H.-J. Kroll and R. Vincenti, PD-sets related to the codes of some classical varieties,, Discrete Math., 301 (2005), 89. doi: 10.1016/j.disc.2004.11.020. Google Scholar [18] F. J. MacWilliams, Permutation decoding of systematic codes,, Bell System Tech. J., 43 (1964), 485. Google Scholar [19] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North-Holland, (1983). Google Scholar [20] R. Peeters, On the $p$-ranks of the adjacency matrices of distance-regular graphs,, J. Algebr. Combin., 15 (2002), 127. doi: 10.1023/A:1013842904024. Google Scholar [21] J. Schönheim, On coverings,, Pacific J. Math., 14 (1964), 1405. Google Scholar
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