On the construction of irreducible and primitive polynomials from [formula omitted] to [formula omitted].

In this paper, we present two constructions of irreducible (primitive) polynomials over F q of degree rm from irreducible (primitive) polynomial over F q m of degree r , and we show that these two constructions coincide. The first construction is based on the Frobenius automorphism of F q m over F q. The second one comes from a generalization of a construction of primitive polynomials over F q which uses the companion matrix. From this generalization, given an irreducible (resp. primitive) polynomial over F q m of degree r , we generate multiple (resp. all) irreducible polynomials over F q of degree rm. As an application, a characterization of the generator polynomial of a BCH code over F q is given. Then, we show how two BCH codes over F q and F q m , respectively, and their generator polynomials are related. [ABSTRACT FROM AUTHOR]