Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility

Abstract

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.