Quasi-Cyclic Codes Via Unfolded Cyclic Codes and Their Reversibility

Abstract

The finite field F q of q elements contains F q as a subfield. If θ ∈ F q is of degree over F q , it can be used to unfold elements of F q to vectors in F q . We apply the unfolding to the coordinates of all codewords of a cyclic code C over F q of length n. This generates a quasi-cyclic code Q over F q of length n and index . 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 C, we present a formula for a generator polynomial matrix for the unfolded code Q. On the other hand, for any quasi-cyclic code Q with a reduced generator polynomial matrix G, we provide a necessary and sufficient condition on G that determines whether or not the code 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 T of the cyclic code 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.INDEX TERMS Cyclotomic cosets, finite fields, generator polynomial matrix, quasi-cyclic codes, reversible codes.

On Reversibility and Self-Duality for Some Classes of Quasi-Cyclic Codes

“…From (5), one concludes that G is diagonal if and only if Q is a direct sum of cyclic codes C i of lengths m. That is…”

“…Several classes of QC codes over F q of length m have been considered in literature. For example, the class of QC codes generated by unfolding cyclic codes of length m over F q [5], [6], and the class of QC codes generated by diagonal generator polynomial matrices, denoted by…”

Good reversible quasi-cyclic codes via unfolding cyclic codes

“…In this paper, we extend our work [2] by presenting some good reversible QC codes in Γ. Here, by a good code, we mean an optimal or suboptimal code, where we say a code is optimal (resp.…”

“…In Theorem 1, we prove a necessary and sufficient condition for the reversibility of QC codes generated by unfolding cyclic codes over F q . Theorem 1 in this paper is more useful than Theorem 3 in [2] because it does not assume a predetermined θ, does not require to build the defining set of the cyclic code, and shows that the self-reciprocity of the minimal polynomial of θ is a sufficient condition for the reversibility but not necessary. As an application of Theorem 1, we present in Table I some good reversible QC codes in Γ of even obtained by computer search.…”

“…The following theorem gives a necessary and sufficient condition for reversibility of QC codes in Γ. Contrary to[2, Theorem 3], Theorem 1 does not require constructing the defining set of the cyclic code C.…”