November  2012, 6(4): 505-516. doi: 10.3934/amc.2012.6.505

Partial permutation decoding for simplex codes

1. 

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

Received  January 2012 Revised  May 2012 Published  November 2012

We show how to find $s$-PD-sets of size $s+1$ that satisfy the Gordon-Schönheim bound for partial permutation decoding for the binary simplex codes $\mathcal S_n(\mathbb F_2)$ for all $n \geq 4$, and for all values of $s$ up to $\left\lfloor\frac{2^n-1}{n}\right\rfloor -1$. The construction also applies to the $q$-ary simplex codes $\mathcal S_n(\mathbb F_q)$ for $q>2$, and to $s$-antiblocking information systems of size $s+1$, for $s$ up to $\left\lfloor\frac{(q^n-1)/(q-1)}{n}\right\rfloor -1$ for all $q$.
Citation: Washiela Fish, Jennifer D. Key, Eric Mwambene. Partial permutation decoding for simplex codes. Advances in Mathematics of Communications, 2012, 6 (4) : 505-516. doi: 10.3934/amc.2012.6.505
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. Symbolic 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, (2006), 3951.   Google Scholar

[4]

D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes,, IEEE Trans. Inform. Theory, 28 (1982), 541.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[5]

E. V. Gorkunov, The group of permutation automorphisms of a $q$-ary Hamming code,, Probl. Inf. Transm., 45 (2009), 309.  doi: 10.1134/S0032946009040024.  Google Scholar

[6]

W. C. Huffman, Codes and groups,, in, (1998), 1345.   Google Scholar

[7]

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

[8]

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

[9]

H.-J. Kroll and R. Vincenti, PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of $PG(5,2)$,, Discrete Math., 308 (2008), 408.  doi: 10.1016/j.disc.2006.11.057.  Google Scholar

[10]

H.-J. Kroll and R. Vincenti, Antiblocking decoding,, Discrete Appl. Math., 158 (2010), 1461.  doi: 10.1016/j.dam.2010.04.007.  Google Scholar

[11]

H.-J. Kroll and R. Vincenti, How to find small AI-systems for antiblocking decoding,, Discrete Math., 312 (2012), 657.  doi: 10.1016/j.disc.2011.06.014.  Google Scholar

[12]

F. J. MacWilliams, Permutation decoding of systematic codes,, Bell System Tech. J., 43 (1964), 485.   Google Scholar

[13]

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

[14]

T. P. McDonough, Private communication,, 2012., ().   Google Scholar

[15]

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

[16]

J. Wolfmann, A permutation decoding of the $(24,12,8)$ Golay code,, IEEE Trans. Inform. Theory, 29 (1983), 748.  doi: 10.1109/TIT.1983.1056726.  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. Symbolic 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, (2006), 3951.   Google Scholar

[4]

D. M. Gordon, Minimal permutation sets for decoding the binary Golay codes,, IEEE Trans. Inform. Theory, 28 (1982), 541.  doi: 10.1109/TIT.1982.1056504.  Google Scholar

[5]

E. V. Gorkunov, The group of permutation automorphisms of a $q$-ary Hamming code,, Probl. Inf. Transm., 45 (2009), 309.  doi: 10.1134/S0032946009040024.  Google Scholar

[6]

W. C. Huffman, Codes and groups,, in, (1998), 1345.   Google Scholar

[7]

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

[8]

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

[9]

H.-J. Kroll and R. Vincenti, PD-sets for binary RM-codes and the codes related to the Klein quadric and to the Schubert variety of $PG(5,2)$,, Discrete Math., 308 (2008), 408.  doi: 10.1016/j.disc.2006.11.057.  Google Scholar

[10]

H.-J. Kroll and R. Vincenti, Antiblocking decoding,, Discrete Appl. Math., 158 (2010), 1461.  doi: 10.1016/j.dam.2010.04.007.  Google Scholar

[11]

H.-J. Kroll and R. Vincenti, How to find small AI-systems for antiblocking decoding,, Discrete Math., 312 (2012), 657.  doi: 10.1016/j.disc.2011.06.014.  Google Scholar

[12]

F. J. MacWilliams, Permutation decoding of systematic codes,, Bell System Tech. J., 43 (1964), 485.   Google Scholar

[13]

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

[14]

T. P. McDonough, Private communication,, 2012., ().   Google Scholar

[15]

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

[16]

J. Wolfmann, A permutation decoding of the $(24,12,8)$ Golay code,, IEEE Trans. Inform. Theory, 29 (1983), 748.  doi: 10.1109/TIT.1983.1056726.  Google Scholar

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