Self-dual 2-quasi-cyclic Codes and Dihedral Codes

TL;DR

This paper characterizes the structure of 2-quasi-cyclic codes over finite fields, establishing conditions under which they are dihedral or self-dual, and explores their equivalences and special cases.

Contribution

It provides a complete characterization of when 2-quasi-cyclic codes are dihedral or self-dual, including necessary and sufficient conditions and their interrelations.

Findings

A characterization of 2-quasi-cyclic codes using Goursat Lemma.

Necessary and sufficient conditions for self-dual 2-quasi-cyclic codes to be dihedral or consta-dihedral.

Equivalence conditions among self-dual double circulant, 2-quasi-cyclic, and dihedral codes.

Abstract

We characterize the structure of 2-quasi-cyclic codes over a finite field F by the so-called Goursat Lemma. With the characterization, we exhibit a necessary and sufficient condition for a 2-quasi-cyclic code being a dihedral code. And we obtain a necessary and sufficient condition for a self-dual 2-quasi-cyclic code being a dihedral code (if charF = 2), or a consta-dihedral code (if charF odd). As a consequence, any self-dual 2-quasi-cyclic code generated by one element must be (consta-)dihedral. In particular, any self-dual double circulant code must be (consta-)dihedral. Also, we show a necessary and sufficient condition that the three classes (the self-dual double circulant codes, the self-dual 2-quasi-cyclic codes, and the self-dual (consta-)dihedral codes) are coincide each other.