# Skew-constacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$

## TL;DR

This paper analyzes the algebraic structure of skew-constacyclic codes over a specific ring, characterizes their automorphisms, and determines conditions for self-duality, advancing coding theory over non-field rings.

## Contribution

It introduces a detailed study of skew-constacyclic codes over the ring rac{}_q[v]/\u2206<v^q - v>, including automorphism groups, generator polynomials, and self-duality conditions.

## Key findings

- Determined the automorphism group of the ring.
- Established generator polynomials for skew-constacyclic codes.
- Provided necessary and sufficient conditions for self-dual codes.

## Abstract

In this paper, the investigation on the algebraic structure of the ring $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ and the description of its automorphism group, enable to study the algebraic structure of codes and their dual over this ring. We explore the algebraic structure of skew-constacyclic codes, by using a linear Gray map and we determine their generator polynomials. Necessary and sufficient conditions for the existence of self-dual skew cyclic and self-dual skew negacyclic codes over $\frac{\mathbb{F}_q[v]}{\langle\,v^q-v\,\rangle}$ are given.