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

Year 2020, , 55 - 71, 29.02.2020
https://doi.org/10.13069/jacodesmath.645026

Abstract

In this work, we define $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G$. We show that the dual of a $G$-code is again a $G$-code in this setting. We study the projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings respectively. We extend known results of constructing $\gamma$-adic codes over $R_\infty$ to $\gamma$-adic $G$-codes over the same ring. We also study $G$-codes over principal ideal rings.

References

  • [1] A. R. Calderbank, N. J. A. Sloane, Modular and $p$–adic cyclic codes, Des. Codes Cryptogr. 6(1) (1995) 21–35.
  • [2] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, 2017.
  • [3] 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) (2018) 2115–2138.
  • [4] S. T. Dougherty, L. Hongwei, Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011) 39–53.
  • [5] S. T. Dougherty, L. Hongwei, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
  • [6] S. T. Dougherty, Y. H. Park, Codes over the $p$–adic integers, Des. Codes and Cryptog. 39(1) (2006) 65–80.
  • [7] H. Horimoto, K. Shiromoto, A Singleton bound for linear codes over quasi–Frobenius rings, Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error– Correcting Codes (1999) 51–52.
  • [8] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
  • [9] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, North–Holland, Amsterdam, 1977.
  • [10] B. R. McDonald, Finite Rings with Identity , New York: Marcel Dekker, Inc, 1974.
  • [11] E. Rains, N. J. A. Sloane, Self–dual codes, in the Handbook of Coding Theory , Pless V.S. and Huffman W.C., eds., Elsevier, Amsterdam, 177–294, 1998.
Year 2020, , 55 - 71, 29.02.2020
https://doi.org/10.13069/jacodesmath.645026

Abstract

References

  • [1] A. R. Calderbank, N. J. A. Sloane, Modular and $p$–adic cyclic codes, Des. Codes Cryptogr. 6(1) (1995) 21–35.
  • [2] S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, 2017.
  • [3] 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) (2018) 2115–2138.
  • [4] S. T. Dougherty, L. Hongwei, Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011) 39–53.
  • [5] S. T. Dougherty, L. Hongwei, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
  • [6] S. T. Dougherty, Y. H. Park, Codes over the $p$–adic integers, Des. Codes and Cryptog. 39(1) (2006) 65–80.
  • [7] H. Horimoto, K. Shiromoto, A Singleton bound for linear codes over quasi–Frobenius rings, Proceedings of the 13th International Symposium on Applied Algebra, Algebraic Algorithms, and Error– Correcting Codes (1999) 51–52.
  • [8] T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
  • [9] F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, North–Holland, Amsterdam, 1977.
  • [10] B. R. McDonald, Finite Rings with Identity , New York: Marcel Dekker, Inc, 1974.
  • [11] E. Rains, N. J. A. Sloane, Self–dual codes, in the Handbook of Coding Theory , Pless V.S. and Huffman W.C., eds., Elsevier, Amsterdam, 177–294, 1998.
There are 11 citations in total.

Details

Primary Language English
Subjects Engineering
Journal Section Articles
Authors

Steven T. Dougherty This is me 0000-0003-4877-1923

Joe Gildea This is me 0000-0001-7242-779X

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

Publication Date February 29, 2020
Published in Issue Year 2020

Cite

APA Dougherty, S. T., Gildea, J., & Korban, A. (2020). $G$-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications, 7(1), 55-71. https://doi.org/10.13069/jacodesmath.645026
AMA Dougherty ST, Gildea J, Korban A. $G$-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. February 2020;7(1):55-71. doi:10.13069/jacodesmath.645026
Chicago Dougherty, Steven T., Joe Gildea, and Adrian Korban. “$G$-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 7, no. 1 (February 2020): 55-71. https://doi.org/10.13069/jacodesmath.645026.
EndNote Dougherty ST, Gildea J, Korban A (February 1, 2020) $G$-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications 7 1 55–71.
IEEE S. T. Dougherty, J. Gildea, and A. Korban, “$G$-codes over formal power series rings and finite chain rings”, Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, pp. 55–71, 2020, doi: 10.13069/jacodesmath.645026.
ISNAD Dougherty, Steven T. et al. “$G$-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications 7/1 (February 2020), 55-71. https://doi.org/10.13069/jacodesmath.645026.
JAMA Dougherty ST, Gildea J, Korban A. $G$-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7:55–71.
MLA Dougherty, Steven T. et al. “$G$-Codes over Formal Power Series Rings and Finite Chain Rings”. Journal of Algebra Combinatorics Discrete Structures and Applications, vol. 7, no. 1, 2020, pp. 55-71, doi:10.13069/jacodesmath.645026.
Vancouver Dougherty ST, Gildea J, Korban A. $G$-codes over formal power series rings and finite chain rings. Journal of Algebra Combinatorics Discrete Structures and Applications. 2020;7(1):55-71.