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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 selforthogonality 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 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.

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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.