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

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

Washiela Fish 1, , Jennifer D. Key 1, and  Eric Mwambene 1,

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.
Keywords: permutation decoding., codes, Uniform subset graphs.
Mathematics Subject Classification: Primary: 05C45, 05B05; Secondary: 94B0.
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
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show all references

References:
[1]

Cambridge University Press, 1992.  Google Scholar

[2]

J. Symb. Comput., 24 (1997), 235-265. doi: 10.1006/jsco.1996.0125.  Google Scholar

[3]

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]

Ph.D thesis, University of KwaZulu-Natal, Durban, 2013. Google Scholar

[5]

Electron. J. Combin., 20 (2013), #P18.  Google Scholar

[6]

Des. Codes Crypt., 68 (2013), 373-393. doi: 10.1007/s10623-011-9594-x.  Google Scholar

[7]

Ph.D thesis, University of the Western Cape, 2007. Google Scholar

[8]

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]

IEEE Trans. Inf. Theory, 28 (1982), 541-543. doi: 10.1109/TIT.1982.1056504.  Google Scholar

[11]

Des. Codes Crypt., 17 (1999), 187-209. doi: 10.1023/A:1008353723204.  Google Scholar

[12]

in Handbook of Coding Theory (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, 1998, 1345-1440.  Google Scholar

[13]

European J. Combin., 26 (2005), 665-682. doi: 10.1016/j.ejc.2004.04.007.  Google Scholar

[14]

Discrete Math., 282 (2004), 171-182. doi: 10.1016/j.disc.2003.12.004.  Google Scholar

[15]

Ars Combin., 79 (2006), 11-19.  Google Scholar

[16]

Discrete Math., 309 (2009), 4663-4681. doi: 10.1016/j.disc.2008.05.032.  Google Scholar

[17]

Discrete Math., 301 (2005), 89-105. doi: 10.1016/j.disc.2004.11.020.  Google Scholar

[18]

Bell System Tech. J., 43 (1964), 485-505. Google Scholar

[19]

North-Holland, Amsterdam, 1983. Google Scholar

[20]

J. Algebr. Combin., 15 (2002), 127-149. doi: 10.1023/A:1013842904024.  Google Scholar

[21]

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

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