American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[2]

S. Bouyuklieva and V. Yorgov, Singly-even codes of length 40, Des. Codes Crypt., 9 (1996), 131-141. doi: 10.1007/BF00124589.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrès Internat. Math., 3 (1970), 211-215.  Google Scholar

[8]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North Holland, Amsterdam, 1977. Google Scholar

[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.  Google Scholar

[10]

E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

S. Bouyuklieva and V. Yorgov, Singly-even codes of length 40, Des. Codes Crypt., 9 (1996), 131-141. doi: 10.1007/BF00124589.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. M. Gleason, Weight polynomials of self-dual codes and the MacWilliams identities, Actes Congrès Internat. Math., 3 (1970), 211-215.  Google Scholar

[8]

F. J. MacWilliams and N. J. A. Sloane, "The Theory of Error-Correcting Codes,'' North Holland, Amsterdam, 1977. Google Scholar

[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.  Google Scholar

[10]

E. M. Rains, Shadow bounds for self-dual-codes, IEEE Trans. Inform. Theory, 44 (1998), 134-139. doi: 10.1109/18.651000.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Suat Karadeniz, Bahattin Yildiz. New extremal binary self-dual codes of length $68$ from $R_2$-lifts of binary self-dual codes. Advances in Mathematics of Communications, 2013, 7 (2) : 219-229. doi: 10.3934/amc.2013.7.219
[2]

Steven T. Dougherty, Joe Gildea, Adrian Korban, Abidin Kaya. Composite constructions of self-dual codes from group rings and new extremal self-dual binary codes of length 68. Advances in Mathematics of Communications, 2020, 14 (4) : 677-702. doi: 10.3934/amc.2020037
[3]

Masaaki Harada, Akihiro Munemasa. On the covering radii of extremal doubly even self-dual codes. Advances in Mathematics of Communications, 2007, 1 (2) : 251-256. doi: 10.3934/amc.2007.1.251
[4]

Stefka Bouyuklieva, Iliya Bouyukliev. Classification of the extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2010, 4 (3) : 433-439. doi: 10.3934/amc.2010.4.433
[5]

Masaaki Harada, Katsushi Waki. New extremal formally self-dual even codes of length 30. Advances in Mathematics of Communications, 2009, 3 (4) : 311-316. doi: 10.3934/amc.2009.3.311
[6]

Gabriele Nebe, Wolfgang Willems. On self-dual MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 633-642. doi: 10.3934/amc.2016031
[7]

Masaaki Harada, Akihiro Munemasa. Classification of self-dual codes of length 36. Advances in Mathematics of Communications, 2012, 6 (2) : 229-235. doi: 10.3934/amc.2012.6.229
[8]

Nikolay Yankov, Damyan Anev, Müberra Gürel. Self-dual codes with an automorphism of order 13. Advances in Mathematics of Communications, 2017, 11 (3) : 635-645. doi: 10.3934/amc.2017047
[9]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[10]

Adrian Korban, Serap Şahinkaya, Deniz Ustun. New type i binary [72, 36, 12] self-dual codes from composite matrices and R1 lifts. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021034
[11]

Steven T. Dougherty, Cristina Fernández-Córdoba, Roger Ten-Valls, Bahattin Yildiz. Quaternary group ring codes: Ranks, kernels and self-dual codes. Advances in Mathematics of Communications, 2020, 14 (2) : 319-332. doi: 10.3934/amc.2020023
[12]

Keita Ishizuka, Ken Saito. Construction for both self-dual codes and LCD codes. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2021070
[13]

Steven T. Dougherty, Joe Gildea, Abidin Kaya, Bahattin Yildiz. New self-dual and formally self-dual codes from group ring constructions. Advances in Mathematics of Communications, 2020, 14 (1) : 11-22. doi: 10.3934/amc.2020002
[14]

Hyun Jin Kim, Heisook Lee, June Bok Lee, Yoonjin Lee. Construction of self-dual codes with an automorphism of order $p$. Advances in Mathematics of Communications, 2011, 5 (1) : 23-36. doi: 10.3934/amc.2011.5.23
[15]

Minjia Shi, Daitao Huang, Lin Sok, Patrick Solé. Double circulant self-dual and LCD codes over Galois rings. Advances in Mathematics of Communications, 2019, 13 (1) : 171-183. doi: 10.3934/amc.2019011
[16]

Bram van Asch, Frans Martens. Lee weight enumerators of self-dual codes and theta functions. Advances in Mathematics of Communications, 2008, 2 (4) : 393-402. doi: 10.3934/amc.2008.2.393
[17]

Bram van Asch, Frans Martens. A note on the minimum Lee distance of certain self-dual modular codes. Advances in Mathematics of Communications, 2012, 6 (1) : 65-68. doi: 10.3934/amc.2012.6.65
[18]

Katherine Morrison. An enumeration of the equivalence classes of self-dual matrix codes. Advances in Mathematics of Communications, 2015, 9 (4) : 415-436. doi: 10.3934/amc.2015.9.415
[19]

Amita Sahni, Poonam Trama Sehgal. Enumeration of self-dual and self-orthogonal negacyclic codes over finite fields. Advances in Mathematics of Communications, 2015, 9 (4) : 437-447. doi: 10.3934/amc.2015.9.437
[20]

Steven T. Dougherty, Cristina Fernández-Córdoba. Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes. Advances in Mathematics of Communications, 2011, 5 (4) : 571-588. doi: 10.3934/amc.2011.5.571

2020 Impact Factor: 0.935

Tools

Metrics

  • PDF downloads (58)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy