On Quasi-Cyclic Codes as a Generalization of Cyclic Codes
Morgan Barbier (INRIA Saclay - Ile de France, LIX), Christophe Chabot, (LJK), Guillaume Quintin (INRIA Saclay - Ile de France, LIX)

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.
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Taxonomy
TopicsCoding theory and cryptography · Rings, Modules, and Algebras · graph theory and CDMA systems
