Dual codes of product semi-linear codes

Let FqFq be a finite field with q   elements and denote by θ:Fq→Fqθ:Fq→Fq an automorphism of FqFq. In this paper, we deal with linear codes of Fqn invariant under a semi-linear map T:Fqn→Fqn for some n≥2n≥2. In particular, we study three kinds of their dual codes, some relations between them and we focus on codes which are products of module skew codes in the non-commutative polynomial ring as a subcase of linear codes invariant by a semi-linear map T. In this setting we give also an algorithm for encoding, decoding and detecting errors and we show a method to construct codes invariant under a fixed T.