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), 235-265. 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, 3951-4023; available at http://magma.maths.usyd.edu.au/magma Google Scholar

[4]

L. Chikamai, Linear Codes Obtained from $2$-Modular Representations of Some Finite Simple Groups, Ph.D thesis, University of KwaZulu-Natal, Durban, 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), #P18.  Google Scholar

[6]

P. Dankelmann, J. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Crypt., 68 (2013), 373-393. doi: 10.1007/s10623-011-9594-x.  Google Scholar

[7]

W. Fish, Codes from Uniform Subset Graphs and Cycle Products, Ph.D thesis, University of the Western Cape, 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), 363-382. 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), 541-543. 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), 187-209. 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), Elsevier, Amsterdam, 1998, 1345-1440.  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), 665-682. 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), 171-182. 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), 11-19.  Google Scholar

[16]

J. D. Key, J. Moori and B. G. Rodrigues, Ternary codes from graphs on triples, Discrete Math., 309 (2009), 4663-4681. 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), 89-105. 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), 485-505. Google Scholar

[19]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1983. Google Scholar

[20]

R. Peeters, On the $p$-ranks of the adjacency matrices of distance-regular graphs, J. Algebr. Combin., 15 (2002), 127-149. doi: 10.1023/A:1013842904024.  Google Scholar

[21]

J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  Google Scholar

show all references

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), 235-265. 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, 3951-4023; available at http://magma.maths.usyd.edu.au/magma Google Scholar

[4]

L. Chikamai, Linear Codes Obtained from $2$-Modular Representations of Some Finite Simple Groups, Ph.D thesis, University of KwaZulu-Natal, Durban, 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), #P18.  Google Scholar

[6]

P. Dankelmann, J. D. Key and B. G. Rodrigues, Codes from incidence matrices of graphs, Des. Codes Crypt., 68 (2013), 373-393. doi: 10.1007/s10623-011-9594-x.  Google Scholar

[7]

W. Fish, Codes from Uniform Subset Graphs and Cycle Products, Ph.D thesis, University of the Western Cape, 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), 363-382. 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), 541-543. 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), 187-209. 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), Elsevier, Amsterdam, 1998, 1345-1440.  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), 665-682. 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), 171-182. 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), 11-19.  Google Scholar

[16]

J. D. Key, J. Moori and B. G. Rodrigues, Ternary codes from graphs on triples, Discrete Math., 309 (2009), 4663-4681. 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), 89-105. 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), 485-505. Google Scholar

[19]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, North-Holland, Amsterdam, 1983. Google Scholar

[20]

R. Peeters, On the $p$-ranks of the adjacency matrices of distance-regular graphs, J. Algebr. Combin., 15 (2002), 127-149. doi: 10.1023/A:1013842904024.  Google Scholar

[21]

J. Schönheim, On coverings, Pacific J. Math., 14 (1964), 1405-1411.  Google Scholar

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