# Self-Dual Skew Cyclic Codes over $\mathbb{F}_{q}+u\mathbb{F}_{q}$

## TL;DR

This paper investigates the existence and construction of Hermitian self-dual skew cyclic and negacyclic codes over a finite chain ring, establishing their properties and connections to skew quasi-twisted codes via Gray maps.

## Contribution

It provides new conditions for the existence of self-dual skew cyclic codes over finite chain rings and extends algorithms for their construction.

## Key findings

- Gray images of codes are equivalent to skew quasi-twisted codes.
- Conditions for Hermitian self-duality over the ring are established.
- An extended algorithm for constructing self-dual codes is proposed.

## Abstract

In this paper, we give conditions for the existence of Hermitian self-dual $\Theta-$cyclic and $\Theta-$negacyclic codes over the finite chain ring $\mathbb{F}_q+u\mathbb{F}_q$. By defining a Gray map from $R=\mathbb{F}_q+u\mathbb{F}_q$ to $\mathbb{F}_{q}^{2}$, we prove that the Gray images of skew cyclic codes of odd length $n$ over $R$ with even characteristic are equivalent to skew quasi-twisted codes of length $2n$ over $\mathbb{F}_q$ of index $2$. We also extend an algorithm of Boucher and Ulmer \cite{BF3} to construct self-dual skew cyclic codes based on the least common left multiples of non-commutative polynomials over $\mathbb{F}_q+u\mathbb{F}_q$.