Notes on Quasi-Cyclic Codes with Cyclic Constituent Codes

Abstract

Quasi-cyclic codes are generalizations of the familiar linear cyclic codes. By using the results of [4], the authors in [2, 3] showed that a quasi-cyclic code \(\mathscr {C}\) over \(\mathbb {F}_q\) of length \(\ell m\) and index \(\ell \) with m being pairwise coprime to \(\ell \) and the characteristic of \(\mathbb {F}_q\) is equivalent to a cyclic code if the constituent codes of \(\mathscr {C}\) are cyclic, where q is a prime power and the equivalence is given in [3]. In this paper, we apply an algebraic method to prove that a quasi-cyclic code with cyclic constituent codes is equivalent to a cyclic code. Moreover, the main result (see Theorem 4) includes Proposition 9 in [3] as a special case.