Abstract
In this paper, we study the class of [Formula: see text]-double cyclic codes of length [Formula: see text] We give a closed formula for the number of [Formula: see text] -double cyclic codes of length [Formula: see text] for any integers [Formula: see text] and [Formula: see text] that are relatively prime to [Formula: see text] Moreover, we give a closed formula for the number of quasi-cyclic (QC) codes of length [Formula: see text] and index [Formula: see text] We also provide formulas for the number of separable and non-separable [Formula: see text]-double cyclic codes of length [Formula: see text] In order to illustrate the results, we calculate the number of some codes with different [Formula: see text] and [Formula: see text]. Moreover, we list optimal parameter [Formula: see text]-double cyclic codes for specific values of [Formula: see text] and [Formula: see text].