G-codes over Formal Power Series Rings and Finite Chain Rings

  • Steven T. Dougherty
  • Joe Gildea
  • Adrian Korban
Keywords: G-codes, Finite chain rings, Formal power series rings, γ-adic codes


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.


A. R. Calderbank, N. J. A. Sloane, Modular and p–adic cyclic codes, Des. Codes Cryptogr. 6(1) (1995) 21–35.

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.