Skew Constacyclic Codes over Finite Chain Rings

TL;DR

This paper extends the theory of skew constacyclic codes to finite chain rings, characterizing their structure, duals, and self-duality conditions, with explicit examples over specific rings.

Contribution

It introduces a comprehensive framework for skew constacyclic codes over finite chain rings, including duals and self-duality, expanding prior work from finite fields and Galois rings.

Findings

Complete structure of skew constacyclic codes over finite chain rings

Explicit formulas for dual and self-dual codes

Illustrative example over +u showing code properties

Abstract

Skew polynomial rings over finite fields ([7] and [10]) and over Galois rings ([8]) have been used to study codes. In this paper, we extend this concept to finite chain rings. Properties of skew constacyclic codes generated by monic right divisors of $x^n-\lambda$, where $\lambda$ is a unit element, are exhibited. When $\lambda^2=1$, the generators of Euclidean and Hermitian dual codes of such codes are determined together with necessary and sufficient conditions for them to be Euclidean and Hermitian self-dual. Of more interest are codes over the ring $\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}$. The structure of all skew constacyclic codes is completely determined. This allows us to express generators of Euclidean and Hermitian dual codes of skew cyclic and skew negacyclic codes in terms of the generators of the original codes. An illustration of all skew cyclic codes of length~2 over $\mathbb{F}_{3}+u\mathbb{F}_{3}$ and their Euclidean and Hermitian dual codes is also provided.