Z₂-double cyclic codes

A binary linear code C is a Z2-double cyclic code if the set of coordinates can be partitioned into two subsets such that any cyclic shift of the coordinates of both subsets leaves invariant the code. These codes can be identified as submodules of the Z2[x]-module Z2[x]/(x^r − 1) × Z2[x]/(x^s − 1). We determine the structure of Z2-double cyclic codes giving the generator polynomials of these codes. The related polynomial representation of Z2-double cyclic codes and its duals, and the relations between the polynomial generators of these codes are studied