Skew cyclic codes over $\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$

TL;DR

This paper explores the structure and properties of skew cyclic codes over a specific ring extension of finite fields, providing generator polynomials, dual code descriptions, and decomposition theorems.

Contribution

It introduces a detailed analysis of skew cyclic codes over the ring $R=F_q+vF_q+v^2F_q$, including generator polynomials and structural properties.

Findings

Describes generator polynomials for skew cyclic codes over $R$

Provides a decomposition theorem for structural analysis

Characterizes dual codes and idempotent generators

Abstract

In this article, we study skew cyclic codes over ring $R=\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}$, where $q=p^{m}$, $p$ is an odd prime and $v^{3}=v$. We describe generator polynomials of skew cyclic codes over this ring and investigate the structural properties of skew cyclic codes over $R$ by a decomposition theorem. We also describe the generator polynomials of the duals of skew cyclic codes. Moreover, the idempotent generators of skew cyclic codes over $\mathbb{F}_{q}$ and $R$ are considered.