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