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

Abstract

In this work, we study two classes of quasi-cyclic (QC) codes and examine how several properties can be combined into the codes of these classes. We start with the class of QC codes generated by diagonal generator polynomial matrices; a QC code in this class is a direct sum of cyclic codes. Then we move on to the class of QC codes of index 2; various binary codes with good parameters are found in this class. In each class, we describe the generator polynomial matrices of reversible codes, self-orthogonal codes, and self-dual codes. Hence, we demonstrate how such properties can be merged in codes of these classes. Particularly for QC codes of index 2, we prove a necessary and sufficient condition for the self-orthogonality of reversible codes. Then we show that reversible QC codes of index 2 are self-dual under the same conditions in which self-dual codes are reversible. We clarify that self-orthogonal reversible QC codes of index 2 over $\mathbb {F}_{q}$ exist for even and odd $q$ , however self-dual reversible codes exist only for even $q$ . Theoretical results are reinforced by several numerical examples. Computer search is used to present some self-dual reversible QC codes of index 2 that have the best known parameters as linear codes. Finally, we highlight the class of 1-generator binary QC codes of index 2 by exploring many self-dual reversible codes that achieve the upper bound on the minimum distance for their parameters.