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Year 2021, , 91 - 105, 20.05.2021
https://doi.org/10.13069/jacodesmath.935951

Abstract

References

  • [1] R. L. Bouzara, K. Guenda, E. Martinez-Moro, Lifted codes and lattices from codes over finite chain rings, arXiv:2007.05871.
  • [2] S. T. Dougherty, Algebraic coding theory over finite commutative rings, SpringerBriefs in Mathematics Springer (2017).
  • [3] S. T. Dougherty, J. Gildea, A. Korban, Extending an established isomorphism between group rings and a subring of the n  n matrices, International Journal of Algebra and Computation, Published: 25 February 2021.
  • [4] S. T. Dougherty, J. Gildea, A. Korban, G- codes over formal power series rings and finite chain rings, J. Algebra Comb. Discrete Appl. 7 (2020) 55–71.
  • [5] S. T. Dougherty, J. Gildea, A. Korban, A. 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 14(4) (2020) 677–702.
  • [6] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, Composite matrices from group rings, composite G-codes and constructions of self-dual codes, arXiv:2002.11614.
  • [7] 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, Cryptogr. 86(9) (2017) 2115–2138.
  • [8] S. T. Dougherty, H. Liu, Y. H. Park, Lifted codes over finite chain rings, Mathematical Journal of Okayama University 53 (2011) 39–53.
  • [9] S. T. Dougherty, H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
  • [10] J. Gildea, A. Kaya, R. Taylor, B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl. 51 (2018) 71–92.
  • [11] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
  • [12] B. R. McDonald, Finite rings with identity, New York: Marcel Dekker (1974).

Composite G-codes over formal power series rings and finite chain rings

Year 2021, , 91 - 105, 20.05.2021
https://doi.org/10.13069/jacodesmath.935951

Abstract

In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.

References

  • [1] R. L. Bouzara, K. Guenda, E. Martinez-Moro, Lifted codes and lattices from codes over finite chain rings, arXiv:2007.05871.
  • [2] S. T. Dougherty, Algebraic coding theory over finite commutative rings, SpringerBriefs in Mathematics Springer (2017).
  • [3] S. T. Dougherty, J. Gildea, A. Korban, Extending an established isomorphism between group rings and a subring of the n  n matrices, International Journal of Algebra and Computation, Published: 25 February 2021.
  • [4] S. T. Dougherty, J. Gildea, A. Korban, G- codes over formal power series rings and finite chain rings, J. Algebra Comb. Discrete Appl. 7 (2020) 55–71.
  • [5] S. T. Dougherty, J. Gildea, A. Korban, A. 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 14(4) (2020) 677–702.
  • [6] S. T. Dougherty, J. Gildea, A. Korban, A. Kaya, Composite matrices from group rings, composite G-codes and constructions of self-dual codes, arXiv:2002.11614.
  • [7] 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, Cryptogr. 86(9) (2017) 2115–2138.
  • [8] S. T. Dougherty, H. Liu, Y. H. Park, Lifted codes over finite chain rings, Mathematical Journal of Okayama University 53 (2011) 39–53.
  • [9] S. T. Dougherty, H. Liu, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
  • [10] J. Gildea, A. Kaya, R. Taylor, B. Yildiz, Constructions for self-dual codes induced from group rings, Finite Fields Appl. 51 (2018) 71–92.
  • [11] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
  • [12] B. R. McDonald, Finite rings with identity, New York: Marcel Dekker (1974).
There are 12 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Adrian Korban This is me 0000-0001-5206-6480

Publication Date May 20, 2021
Published in Issue Year 2021

Cite

APA Korban, A. (2021). Composite G-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 8(2), 91-105. https://doi.org/10.13069/jacodesmath.935951
AMA Korban A. Composite G-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. May 2021;8(2):91-105. doi:10.13069/jacodesmath.935951
Chicago Korban, Adrian. “Composite G-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 8, no. 2 (May 2021): 91-105. https://doi.org/10.13069/jacodesmath.935951.
EndNote Korban A (May 1, 2021) Composite G-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications 8 2 91–105.
IEEE A. Korban, “Composite G-codes over formal power series rings and finite chain rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 2, pp. 91–105, 2021, doi: 10.13069/jacodesmath.935951.
ISNAD Korban, Adrian. “Composite G-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 8/2 (May 2021), 91-105. https://doi.org/10.13069/jacodesmath.935951.
JAMA Korban A. Composite G-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8:91–105.
MLA Korban, Adrian. “Composite G-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 8, no. 2, 2021, pp. 91-105, doi:10.13069/jacodesmath.935951.
Vancouver Korban A. Composite G-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2021;8(2):91-105.