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

Year 2021, Volume: 8 Issue: 2, 91 - 105, 20.05.2021

Adrian Korban

https://doi.org/10.13069/jacodesmath.935951

Abstract

In this paper, we extend the work done on $G$-codes over formal power series rings and finite chain rings $\mathbb{F}_q[t]/(t^i)$, to composite $G$-codes over the same alphabets. We define composite $G$-codes over the infinite ring $R_\infty$ as ideals in the group ring $R_\infty G.$ We show that the dual of a composite $G$-code is again a composite $G$-code in this setting. We extend the known results on projections and lifts of $G$-codes over the finite chain rings and over the formal power series rings to composite $G$-codes. Additionally, we extend some known results on $\gamma$-adic $G$-codes over $R_\infty$ to composite $G$-codes and study these codes over principal ideal rings.

Keywords

Composite $G$-codes, Group rings, Finite chain rings, Formal power series rings, $p$-adic integers

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