Skew cyclic codes over $\mathbb{F}_{p}+u\mathbb{F}_{p}$

TL;DR

This paper investigates skew cyclic codes over the ring _{p}+u_{p} with odd prime p, characterizing their structure, providing generator polynomials, and presenting encoding/decoding algorithms, including new optimal and quasi-twisted codes.

Contribution

It offers a comprehensive characterization of skew cyclic codes over _{p}+u_{p}, including generator polynomials, minimal spanning sets, and algorithms, along with new optimal codes and a novel ternary quasi-twisted code.

Findings

Characterized all skew cyclic codes as left modules.

Provided explicit generator polynomials and minimal spanning sets.

Constructed new optimal codes and a novel ternary quasi-twisted code.

Abstract

In this paper, we study skew cyclic codes with arbitrary length over the ring $R=\mathbb{F}_{p}+u\mathbb{F}_{p}$ where $p$ is an odd prime and $% u^{2}=0$. We characterize all skew cyclic codes of length $n$ as left $% R[x;\theta ]$-submodules of $R_{n}=R[x;\theta ]/\langle x^{n}-1\rangle $. We find all generator polynomials for these codes and describe their minimal spanning sets. Moreover, an encoding and decoding algorithm is presented for skew cyclic codes over the ring $R$. Finally, based on the theory we developed in this paper, we provide examples of codes with good parameters over $F_{p}$ with different odd prime $p.$ In fact, example 25 in our paper is a new ternary code in the class of quasi-twisted codes. The other examples we provided are examples of optimal codes.