On the Construction of Skew Quasi-Cyclic Codes

TL;DR

This paper introduces skew quasi-cyclic codes constructed over non-commutative skew polynomial rings, providing new code constructions with improved Hamming distances compared to existing codes.

Contribution

It develops the theory of skew QC codes, including generator and parity-check polynomials, and demonstrates the construction of codes with superior Hamming distances.

Findings

Constructed new skew QC codes with improved Hamming distances

Established the uniqueness of parity-check polynomials up to similarity

Provided a framework for code construction using skew polynomial rings

Abstract

In this paper we study a special type of quasi-cyclic (QC) codes called skew QC codes. This set of codes is constructed using a non-commutative ring called the skew polynomial rings $F[x;\theta ]$. After a brief description of the skew polynomial ring $F[x;\theta ]$ it is shown that skew QC codes are left submodules of the ring $R_{s}^{l}=(F[x;\theta ]/(x^{s}-1))^{l}.$ The notions of generator and parity-check polynomials are given. We also introduce the notion of similar polynomials in the ring $F[x;\theta ]$ and show that parity-check polynomials for skew QC codes are unique up to similarity. Our search results lead to the construction of several new codes with Hamming distances exceeding the Hamming distances of the previously best known linear codes with comparable parameters.