Binary quasi-cyclic codes of index 2 and skew polynomial rings

We present a study of the factorization of the polynomial Xm−1 in M2(F2)[X] and we determine the period of any reversible polynomial of this polynomial ring by using skew polynomial rings. These results are applied to the construction of the class of quasi-cyclic codes Ω(P) introduced by Cayrel et al. Furthermore, we present a new construction of the self dual subclass.