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

Year 2020, , 103 - 119, 07.05.2020

Joe Gıldea Abidin Kaya Alexander Tylyshchak Bahattin Yıldız

https://doi.org/10.13069/jacodesmath.729402

Cited By: 5

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.

Keywords

Group rings, Self-dual codes, Codes over rings, Bordered constructions

References

• [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.