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New extremal binary self-dual codes of length 68

Year 2014, Volume: 1 Issue: 1, 29 - 39, 01.03.2014
https://doi.org/10.13069/jacodesmath.79879

Abstract

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F_2 + uF_2 + u^2F_2, where u^3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with beta=117, 120, 133 in W68;1 and with gamma = 1, beta=49, 57, 59 and codes with gamma=2, beta=69, 81 in W68;2 are constructed for the ?first time in the literature. In addition to these, some known such codes are reconstructed via this extension. The results are tabulated.

References

  • J. H. Conway, N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36(6), 1319-1333, 1990.
  • S. T. Dougherty, T. A. Gulliver, M., Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43(6), 2036-2047, 1997.
  • P. Gaborit, Quadratic double circulant codes over fields, Journal of Combinatorial Theory Series A, 97(1), 85-107, 2002.
  • W.C. Huffman, V. Pless, Fundamentals of error correcting codes, Cambridge University press, 2003.
  • J.-L. Kim, New extremal self-dual codes of lengths 36, 38 and 58, IEEE Trans. Inf. Theory, 47(1), 386-393, 2001.
  • A. Kaya, B. Yildiz, İ. Şiap, New extremal binary self-dual codes from F4+ uF4-lifts of quadratic double circulant codes over F, available online at http://arxiv.org/abs/1405.7147
  • A. Kaya, B. Yildiz, Binary generator matrices of new extremal self-dual binary codes of length 68, available online at http://www.fatih.edu.tr/~akaya/binary/68u31.txt
  • A. Kaya, B. Yildiz, Extension theorems for self-dual codes over rings and new binary self-dual codes, available online at http://arxiv.org/abs/1404.0195.
  • E. M. Rains, Shadow Bounds for Self Dual Codes, IEEE Trans. Inf. Theory, 44(1), 134-139, 1998.
  • H.-P. Tsai, P.-Y. Shih, R.-Y. Wuh, W.-K. Su, C.-H. Chen, Construction of self-dual codes, IEEE
  • Trans. Inform. Theory, 54(8), 3826-3831, 2008.

New extremal binary self-dual codes of length 68

Year 2014, Volume: 1 Issue: 1, 29 - 39, 01.03.2014
https://doi.org/10.13069/jacodesmath.79879

Abstract

In this correspondence, we consider quadratic double and bordered double circulant construction methods over the ring R := F_2 + uF_2 + u^2F_2, where u^3 = 1. Among other examples, extremal binary self-dual codes of length 66 are obtained by these constructions. These are extended by using extension theorems for self-dual codes and as a result 8 new extremal binary self-dual codes of length 68 are obtained. More precisely, codes with beta=117, 120, 133 in W68;1 and with gamma = 1, beta=49, 57, 59 and codes with gamma=2, beta=69, 81 in W68;2 are constructed for the ?first time in the literature. In addition to these, some known such codes are reconstructed via this extension. The results are tabulated.

References

  • J. H. Conway, N. J. A. Sloane, A new upper bound on the minimal distance of self-dual codes, IEEE Trans. Inform. Theory, 36(6), 1319-1333, 1990.
  • S. T. Dougherty, T. A. Gulliver, M., Harada, Extremal binary self dual codes, IEEE Trans. Inform. Theory, 43(6), 2036-2047, 1997.
  • P. Gaborit, Quadratic double circulant codes over fields, Journal of Combinatorial Theory Series A, 97(1), 85-107, 2002.
  • W.C. Huffman, V. Pless, Fundamentals of error correcting codes, Cambridge University press, 2003.
  • J.-L. Kim, New extremal self-dual codes of lengths 36, 38 and 58, IEEE Trans. Inf. Theory, 47(1), 386-393, 2001.
  • A. Kaya, B. Yildiz, İ. Şiap, New extremal binary self-dual codes from F4+ uF4-lifts of quadratic double circulant codes over F, available online at http://arxiv.org/abs/1405.7147
  • A. Kaya, B. Yildiz, Binary generator matrices of new extremal self-dual binary codes of length 68, available online at http://www.fatih.edu.tr/~akaya/binary/68u31.txt
  • A. Kaya, B. Yildiz, Extension theorems for self-dual codes over rings and new binary self-dual codes, available online at http://arxiv.org/abs/1404.0195.
  • E. M. Rains, Shadow Bounds for Self Dual Codes, IEEE Trans. Inf. Theory, 44(1), 134-139, 1998.
  • H.-P. Tsai, P.-Y. Shih, R.-Y. Wuh, W.-K. Su, C.-H. Chen, Construction of self-dual codes, IEEE
  • Trans. Inform. Theory, 54(8), 3826-3831, 2008.
There are 11 citations in total.

Details

Primary Language English
Journal Section Articles
Authors

Abidin Kaya This is me

Bahattin Yildiz

Publication Date March 1, 2014
Published in Issue Year 2014 Volume: 1 Issue: 1

Cite

APA Kaya, A., & Yildiz, B. (2014). New extremal binary self-dual codes of length 68. Journal of Algebra Combinatorics Discrete Structures and Applications, 1(1), 29-39. https://doi.org/10.13069/jacodesmath.79879
AMA Kaya A, Yildiz B. New extremal binary self-dual codes of length 68. Journal of Algebra Combinatorics Discrete Structures and Applications. March 2014;1(1):29-39. doi:10.13069/jacodesmath.79879
Chicago Kaya, Abidin, and Bahattin Yildiz. “New Extremal Binary Self-Dual Codes of Length 68”. Journal of Algebra Combinatorics Discrete Structures and Applications 1, no. 1 (March 2014): 29-39. https://doi.org/10.13069/jacodesmath.79879.
EndNote Kaya A, Yildiz B (March 1, 2014) New extremal binary self-dual codes of length 68. Journal of Algebra Combinatorics Discrete Structures and Applications 1 1 29–39.
IEEE A. Kaya and B. Yildiz, “New extremal binary self-dual codes of length 68”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 1, no. 1, pp. 29–39, 2014, doi: 10.13069/jacodesmath.79879.
ISNAD Kaya, Abidin - Yildiz, Bahattin. “New Extremal Binary Self-Dual Codes of Length 68”. Journal of Algebra Combinatorics Discrete Structures and Applications 1/1 (March 2014), 29-39. https://doi.org/10.13069/jacodesmath.79879.
JAMA Kaya A, Yildiz B. New extremal binary self-dual codes of length 68. Journal of Algebra Combinatorics Discrete Structures and Applications. 2014;1:29–39.
MLA Kaya, Abidin and Bahattin Yildiz. “New Extremal Binary Self-Dual Codes of Length 68”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 1, no. 1, 2014, pp. 29-39, doi:10.13069/jacodesmath.79879.
Vancouver Kaya A, Yildiz B. New extremal binary self-dual codes of length 68. Journal of Algebra Combinatorics Discrete Structures and Applications. 2014;1(1):29-3.