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Journals / Journal of Algebra Combinatorics Discrete Structures and Applications / 2026 / Cilt: 7 Sayı: 2

A modified bordered construction for self-dual codes from group rings

Joe Gıldea(University of Chester)
Abidin Kaya(Sampoerna University)0000-0003-0175-1909
Alexander Tylyshchak(Uzhgorod State University)
Bahattin Yıldız(Northern Arizona University)0000-0001-8106-3123
Pages
103–119
Publication Date
2020-05-07
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

In particular we find a new extremal binary self-dual code of length 78.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We describe a bordered construction for self-dual codes coming from group rings.
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Key Findings & Data:We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths.
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Conclusion & Impact:In particular we find a new extremal binary self-dual code of length 78.
🔬Design:Araştırma Makalesi
📍Scope:Comprehensive Sample
⚙️Variables:Group rings · Self-dual codes · Codes over rings

Abstract

We describe a bordered construction for self-dual codes coming from group rings. We apply the constructions coming from the cyclic and dihedral groups over several alphabets to obtain extremal binary self-dual codes of various lengths. In particular we find a new extremal binary self-dual code of length 78.

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Cite This Article

Gıldea, J., Kaya, A., Tylyshchak, A., Yıldız, B. (2020). A modified bordered construction for self-dual codes from group rings. Journal of Algebra Combinatorics Discrete Structures and Applications, Cilt: 7 Sayı: 2, 103–119. https://doi.org/10.13069/jacodesmath.729402

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References (22)

  1. D. Anev, N. Yankov, Self–dual codes of length 78 with an automorphism of order 13, in XVth International Workshop on Optimal Codes and Related Topics, Sofia, Bulgaria, July 2017.
  2. F. Bernhardt, P. Landrock, O. Manz, The extended Golay codes considered as ideals, J. Combin. Theory Ser. A 55(2) (1990) 235–246.
  3. 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) (1990) 1319–1333.
  4. R. Dontcheva, New binary [70; 35; 12] self-dual and binary [72; 36; 12] self-dual doubly-even codes, Serdica Mathematical Journal 24(4) (2001) 287–302.
  5. S. T. Dougherty, J. Gildea, R. Taylor, A. Tylshchak, Group rings, G-Codes and constructions of self-dual and formally self-dual codes, Des. Codes Crypt. 86(9) (2018) 2115–2138.
  6. S. T. Dougherty, P. Gaborit, M. Harada, P. Sole, Type II codes over $\mathbb{F}_{2}+u\mathbb{F}_{2}$, IEEE Trans. Inform. Theory 45 (1999) 32–45.
  7. S. T. Dougherty, M. Harda, T. A. Gulliver, Extremal binary self-dual codes, IEEE Trans. Inform. Theory 43(6) (1997) 2036–2047.
  8. S. T. Dougherty, B. Yildiz, S. Karadeniz, Codes over $R_k$, Gray maps and their binary images, Finite Fields Appl. 17(3) (2011) 205–219.
  9. S. T. Dougherty, B. Yildiz, S. Karadeniz, Self–dual codes over $R_k$ and binary self-dual codes, Eur. J. Pure Appl. Mathematics 6(1) (2013) 89–106.
  10. P. Gaborit, V. Pless, P. Sole, O. Atkin, Type II codes over $\mathbb{F}_{4}$, Finite Fields Appl. 8(2) (2002) 171–183.
  11. J. Gildea, A. Kaya, R. Taylor, B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl. 51 (2018) 71–92.
  12. M. Gurel, N. Yankov, Self–dual codes with an automophism of order 17, Mathematical Commun. 21(1) (2016) 97–107.
  13. M. Harada, The existence of a self–dual [70; 35; 12] code and formally self–dual codes, Finite Fields Appl. 3(2) (1997) 131–139.
  14. M. Harada, K. Saito, Singly even self–dual codes constructed from Hadamard matrices of order 28, Australas. J. Combin. 70 (2018) 288–296.
  15. T. Hurley, Group rings and rings of matrices, Int. J. Pure and Appl. Math. 31(3) (2006) 319–335.
  16. T. Hurley, Self–dual, dual–containing and related quantum codes from group rings, arXiv:0711.3983, 2007.
  17. S. Ling, P. Sole, Type II codes over $\mathbb{F}_{4}+u\mathbb{F}_{4}$", Europ. J. Combinatorics 22 (2001) 983–997.
  18. I. McLoughlin, A group ring construction of the [48; 24; 12] Type II linear block code, Des. Codes Crypt. 63(1) (2012) 29–41.
  19. I. McLoughlin, T. Hurley, A group ring construction of the extended binary Golay code, IEEE Trans. Inform. Theory 54(9) (2008) 4381–4383.
  20. E. M. Rains, Shadow bounds for self–dual codes, IEEE Trans. Inform. Theory 44 (1998) 134–139.
  21. N. Yankov, D. Anev, M. Gurel, Self-dual codes with an automorphism of order 13, Adv. Math. Commun. 11(3) (2017) 635–645.
  22. T. Zhang, J. Michel, T. Feng, G. Ge, On the existence of certain optimal self–dual codes with lengths between 74 and 116, The Electronic Journal of Combinatorics 22(4) (2015) 1–25.