On the performance of binary extremal self-dual codes

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May 2011, 5(2): 267-274. doi: 10.3934/amc.2011.5.267

On the performance of binary extremal self-dual codes

Stefka Bouyuklieva ^1, , Anton Malevich ^2, and Wolfgang Willems ^2,

1.	Department of Mathematics and Informatics, Veliko Tarnovo University, 5000 Veliko Tarnovo, Bulgaria
2.	Department of Mathematics, Otto-von-Guericke-University, 39016 Magdeburg, Germany, Germany

Received April 2010 Revised July 2010 Published May 2011

Abstract
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The decoding error probability of a code $C$ measures the quality of performance when $C$ is used for error correction in data transmission. In this note we compare different types of codes with regard to the decoding error probability.

Keywords: decoding error probabilities., Binary self-dual extremal codes.

Mathematics Subject Classification: Primary: 94B05, 94B70; Secondary: 94B3.

Citation: Stefka Bouyuklieva, Anton Malevich, Wolfgang Willems. On the performance of binary extremal self-dual codes. Advances in Mathematics of Communications, 2011, 5 (2) : 267-274. doi: 10.3934/amc.2011.5.267

References:

[1]	C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory A, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.
[2]	S. Bouyuklieva and V. Yorgov, Singly-even codes of length 40, Des. Codes Crypt., 9 (1996), 131-141. doi: 10.1007/BF00124589.
[3]	Y. Cheng and N. J. A. Sloane, Codes from symmetry groups, and a $[32,17,8]$ code, SIAM J. Discrete Math., 2 (1989), 28-37. doi: 10.1137/0402003.
[4]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[5]	R. Doncheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Appl. Algebra Eng. Comm. Comp., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.
[6]	A. Faldum, J. Lafuente, G. Ochoa and W. Willems, Error probabilities for bounded distance decoding, Des. Codes Crypt., 40 (2006), 237-252. doi: 10.1007/s10623-006-0010-x.
[7]	A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrès Internat. Math., 3 (1970), 211-215.
[8]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North Holland, Amsterdam, 1977.
[9]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. doi: 10.1016/S0019-9958(73)90273-8.
[10]	E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[11]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.
[12]	V. Yorgov, On the minimal weight of some singly-even codes, IEEE Trans. Inform. Theory, 45 (1999), 2539-2541. doi: 10.1109/18.796401.
[13]	S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286. doi: 10.1016/S0166-218X(98)00131-0.

show all references

References:

[1]	C. Bachoc and P. Gaborit, Designs and self-dual codes with long shadows, J. Combin. Theory A, 105 (2004), 15-34. doi: 10.1016/j.jcta.2003.09.003.
[2]	S. Bouyuklieva and V. Yorgov, Singly-even codes of length 40, Des. Codes Crypt., 9 (1996), 131-141. doi: 10.1007/BF00124589.
[3]	Y. Cheng and N. J. A. Sloane, Codes from symmetry groups, and a $[32,17,8]$ code, SIAM J. Discrete Math., 2 (1989), 28-37. doi: 10.1137/0402003.
[4]	J. H. Conway and N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36 (1990), 1319-1333. doi: 10.1109/18.59931.
[5]	R. Doncheva and M. Harada, Some extremal self-dual codes with an automorphism of order 7, Appl. Algebra Eng. Comm. Comp., 14 (2003), 75-79. doi: 10.1007/s00200-003-0126-4.
[6]	A. Faldum, J. Lafuente, G. Ochoa and W. Willems, Error probabilities for bounded distance decoding, Des. Codes Crypt., 40 (2006), 237-252. doi: 10.1007/s10623-006-0010-x.
[7]	A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrès Internat. Math., 3 (1970), 211-215.
[8]	F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North Holland, Amsterdam, 1977.
[9]	C. L. Mallows and N. J. A. Sloane, An upper bound for self-dual codes, Inform. Control, 22 (1973), 188-200. doi: 10.1016/S0019-9958(73)90273-8.
[10]	E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.
[11]	E. M. Rains and N. J. A. Sloane, Self-dual codes, in "Handbook of Coding Theory'' (eds. V.S. Pless and W.C. Huffman), Elsevier, Amsterdam, (1998), 177-294.
[12]	V. Yorgov, On the minimal weight of some singly-even codes, IEEE Trans. Inform. Theory, 45 (1999), 2539-2541. doi: 10.1109/18.796401.
[13]	S. Zhang, On the nonexistence of extremal self-dual codes, Discrete Appl. Math., 91 (1999), 277-286. doi: 10.1016/S0166-218X(98)00131-0.

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