Year 2020, Volume 7 , Issue 2, Pages 103 - 119 2020-05-07

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

Joe GILDEA [1] , Abidin KAYA [2] , Alexander TYLYSHCHAK [3] , Bahattin YILDIZ [4]


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.
Group rings, Self-dual codes, Codes over rings, Bordered constructions
  • [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.
Primary Language en
Subjects Engineering
Journal Section Articles
Authors

Author: Joe GILDEA
Institution: University of Chester
Country: United Kingdom


Orcid: 0000-0003-0175-1909
Author: Abidin KAYA
Institution: Sampoerna University
Country: Indonesia


Author: Alexander TYLYSHCHAK
Institution: Uzhgorod State University
Country: Ukraine


Orcid: 0000-0001-8106-3123
Author: Bahattin YILDIZ
Institution: Northern Arizona University
Country: United States


Dates

Publication Date : May 7, 2020

Bibtex @research article { jacodesmath729402, journal = {Journal of Algebra Combinatorics Discrete Structures and Applications}, issn = {}, eissn = {2148-838X}, address = {}, publisher = {Yildiz Technical University}, year = {2020}, volume = {7}, pages = {103 - 119}, doi = {10.13069/jacodesmath.729402}, title = {A modified bordered construction for self-dual codes from group rings}, key = {cite}, author = {Gıldea, Joe and Kaya, Abidin and Tylyshchak, Alexander and Yıldız, Bahattin} }
APA 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 , 7 (2) , 103-119 . DOI: 10.13069/jacodesmath.729402
MLA Gıldea, J , Kaya, A , Tylyshchak, A , Yıldız, B . "A modified bordered construction for self-dual codes from group rings" . Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 103-119 <https://dergipark.org.tr/en/pub/jacodesmath/issue/54201/729402>
Chicago Gıldea, J , Kaya, A , Tylyshchak, A , Yıldız, B . "A modified bordered construction for self-dual codes from group rings". Journal of Algebra Combinatorics Discrete Structures and Applications 7 (2020 ): 103-119
RIS TY - JOUR T1 - A modified bordered construction for self-dual codes from group rings AU - Joe Gıldea , Abidin Kaya , Alexander Tylyshchak , Bahattin Yıldız Y1 - 2020 PY - 2020 N1 - doi: 10.13069/jacodesmath.729402 DO - 10.13069/jacodesmath.729402 T2 - Journal of Algebra Combinatorics Discrete Structures and Applications JF - Journal JO - JOR SP - 103 EP - 119 VL - 7 IS - 2 SN - -2148-838X M3 - doi: 10.13069/jacodesmath.729402 UR - https://doi.org/10.13069/jacodesmath.729402 Y2 - 2019 ER -
EndNote %0 Journal of Algebra Combinatorics Discrete Structures and Applications A modified bordered construction for self-dual codes from group rings %A Joe Gıldea , Abidin Kaya , Alexander Tylyshchak , Bahattin Yıldız %T A modified bordered construction for self-dual codes from group rings %D 2020 %J Journal of Algebra Combinatorics Discrete Structures and Applications %P -2148-838X %V 7 %N 2 %R doi: 10.13069/jacodesmath.729402 %U 10.13069/jacodesmath.729402
ISNAD Gıldea, Joe , Kaya, Abidin , Tylyshchak, Alexander , Yıldız, Bahattin . "A modified bordered construction for self-dual codes from group rings". Journal of Algebra Combinatorics Discrete Structures and Applications 7 / 2 (May 2020): 103-119 . https://doi.org/10.13069/jacodesmath.729402
AMA Gıldea J , Kaya A , Tylyshchak A , Yıldız B . A modified bordered construction for self-dual codes from group rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(2): 103-119.
Vancouver Gıldea J , Kaya A , Tylyshchak A , Yıldız B . A modified bordered construction for self-dual codes from group rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020; 7(2): 103-119.