$\mathbb{Z}_{q}(\mathbb{Z}_{q}+u\mathbb{Z}_{q})$-Linear Skew Constacyclic Codes

TL;DR

This paper investigates skew constacyclic codes over a specific ring extension involving $ ext{Z}_q$ and nilpotent elements, providing structural insights, generator descriptions, and new code constructions over $ ext{Z}_4$.

Contribution

It introduces the concept of skew constacyclic codes over the ring $ ext{Z}_q( ext{Z}_q + u ext{Z}_q)$, explores their algebraic structure, and constructs new linear codes via Gray images.

Findings

Described generator polynomials for these codes.

Constructed new linear codes over $ ext{Z}_4$ using Gray images.

Generalized codes to double skew constacyclic codes.

Abstract

In this paper, we study skew constacyclic codes over the ring $\mathbb{Z}_{q}R$ where $R=\mathbb{Z}_{q}+u\mathbb{Z}_{q}$, $q=p^{s}$ for a prime $p$ and $u^{2}=0$. We give the definition of these codes as subsets of the ring $\mathbb{Z}_{q}^{\alpha}R^{\beta}$. Some structural properties of the skew polynomial ring $ R[x,\theta]$ are discussed, where $ \theta$ is an automorphism of $R$. We describe the generator polynomials of skew constacyclic codes over $ R $ and $\mathbb{Z}_{q}R$. Using Gray images of skew constacyclic codes over $\mathbb{Z}_{q}R$ we obtained some new linear codes over $\mathbb{Z}_4$. Further, we have generalized these codes to double skew constacyclic codes over $\mathbb{Z}_{q}R$.