$G$-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

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