Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility

Résumé

The finite field$\mathbb {F}_{q^\ell }$of$q^\ell $elements contains$\mathbb {F}_{q}$as a subfield. If$\theta \in \mathbb {F}_{q^\ell }$is of degree$\ell $over$\mathbb {F}_{q}$, it can be used to unfold elements of$\mathbb {F}_{q^\ell }$to vectors in$\mathbb {F}_{q}^\ell $. We apply the unfolding to the coordinates of all codewords of a cyclic code$\mathcal {C}$over$\mathbb {F}_{q^\ell }$of length$n$. This generates a quasi-cyclic code$\mathcal {Q}$over$\mathbb {F}_{q}$of length$n\ell $and index$\ell $. We focus on the class of quasi-cyclic codes resulting from the unfolding of cyclic codes. Given a generator polynomial$g(x)$of a cyclic code$\mathcal {C}$, we present a formula for a generator polynomial matrix for the unfolded code$\mathcal {Q}$. On the other hand, for any quasi-cyclic code$\mathcal {Q}$with a reduced generator polynomial matrix$G$, we provide a necessary and sufficient condition on$G$that determines whether or not the code$\mathcal {Q}$can be represented as the unfolding of a cyclic code. Furthermore, as an application, we discuss the reversibility of the class of quasi-cyclic codes resulting from unfolding of cyclic codes. Specifically, we provide a necessary and sufficient condition on the defining set$\mathcal {T}$of the cyclic code$\mathcal {C}$that ensures the reversibility of the unfolded code. Numerical examples are used to illustrate theoretical results. Some of these examples show that quasi-cyclic codes reversibility does not necessarily require a self-reciprocal generator polynomial for the cyclic code. Since reversibility is essential in constructing DNA codes, some DNA codes are designed as examples.

Citer ce document

Accès au document

Ce document est hébergé sur son dépôt institutionnel d'origine.

Auteur(s)

Statistiques

Consultations : 1

Téléchargements : 0