G-codes over Formal Power Series Rings and Finite Chain Rings
Keywords:
G-codes, Finite chain rings, Formal power series rings, γ-adic codes
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
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S. T. Dougherty, Algebraic Coding Theory over Finite Commutative Rings, Springer Briefs in Mathematics, Springer, 2017.
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.
S. T. Dougherty, L. Hongwei, Y. H. Park, Lifted codes over finite chain rings, Math. J. Okayama Univ. 53 (2011) 39–53.
S. T. Dougherty, L. Hongwei, Cyclic codes over formal power series rings, Acta Mathematica Scientia 31(1) (2011) 331–343.
S. T. Dougherty, Y. H. Park, Codes over the p–adic integers, Des. Codes and Cryptog. 39(1) (2006) 65–80.
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.
T. Hurley, Group rings and rings of matrices, Int. Jour. Pure and Appl. Math 31(3) (2006) 319–335.
F. J. MacWilliams, N. J. A. Sloane, The Theory of Error–Correcting Codes, North–Holland, Amsterdam, 1977.
B. R. McDonald, Finite Rings with Identity , New York: Marcel Dekker, Inc, 1974.
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.
Published
2020-01-15
