$${{mathbb {Z}}}_2$$-double cyclic codes

A binary linear code C is a ({mathbb {Z}}_2)-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 ({mathbb {Z}}_2[x])-module ({mathbb {Z}}_2[x]/(x^r-1)times {mathbb {Z}}_2[x]/(x^s-1).) We determine the structure of ({mathbb {Z}}_2)-double cyclic codes giving the generator polynomials of these codes. We give the polynomial representation of ({mathbb {Z}}_2)-double cyclic codes and its duals, and the relations between the generator polynomials of these codes. Finally, we study the relations between ({{mathbb {Z}}}_2)-double cyclic and other families of cyclic codes, and show some examples of distance optimal ({mathbb {Z}}_2)-double cyclic codes.