On Reversible Z2-Double Cyclic Codes

Abstract

A binary linear code is said to be a ℤ2-double cyclic code if its coordinates can be partitioned into two subsets such that any simultaneous cyclic shift of the coordinates of the subsets leaves the code invariant. These codes were introduced in [6]. A ℤ2-double cyclic code is called reversible if reversing the order of the coordinates of the two subsets leaves the code invariant. In this note, we give necessary and sufficient conditions for a ℤ2-double cyclic code to be reversible. We also give a relation between reversible ℤ2-double cyclic code and LCD ℤ2-double cyclic code for the separable case and we present a few examples to show that such a relation doesn't hold in the non-separable case. Furthermore, we list examples of reversible ℤ2-double cyclic codes of length ≤ 10.