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

Year 2020, Volume: 7 Issue: 1 (Special Issue in Algebraic Coding Theory: New Trends and Its Connections), 55 - 71, 29.02.2020

Steven T. Dougherty Joe Gildea Adrian Korban

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.

Keywords

$G$-codes , Finite chain rings , Formal power series rings , $\gamma$-adic codes

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.