On Quasi-Cyclic Codes as a Generalization of Cyclic Codes

TL;DR

This paper explores quasi-cyclic codes as a generalization of cyclic codes, establishing their algebraic structure, constructing new code classes, and proposing decoding algorithms, leading to codes with improved parameters.

Contribution

It introduces a new algebraic framework for quasi-cyclic codes, generalizes properties of cyclic codes, and constructs novel codes with better parameters than existing ones.

Findings

Established a one-to-one correspondence between quasi-cyclic codes and ideals in polynomial rings.

Constructed new classes of quasi-BCH and quasi-evaluation codes.

Discovered a new quasi-cyclic code with superior parameters and derived codes surpassing known bounds.

Abstract

In this article we see quasi-cyclic codes as block cyclic codes. We generalize some properties of cyclic codes to quasi-cyclic ones such as generator polynomials and ideals. Indeed we show a one-to-one correspondence between l-quasi-cyclic codes of length m and ideals of M_l(Fq)[X]/(X^m-1). This permits to construct new classes of codes, namely quasi-BCH and quasi-evaluation codes. We study the parameters of such codes and propose a decoding algorithm up to half the designed minimum distance. We even found one new quasi-cyclic code with better parameters than known [189, 11, 125]_F4 and 48 derivated codes beating the known bounds as well.