A Note on the Dual Codes of Module Skew Codes

In [4], starting from an automorphism θ of a finite field Fq and a skew polynomial ring R = Fq[X; θ], module θ-codes are defined as left R-submodules of R/Rf where f ∈ R. In [4] it is conjectured that an Euclidean self-dual module θ-code is a θ-constacyclic code and a proof is given in the special case when the order of θ divides the length of the code. In this paper we prove that this conjecture holds in general by showing that the dual of a module θ-code is a module θ-code if and only if it is a θ-constacyclic code. Furthermore, we establish that a module θ-code which is not θ-constacyclic is a shortened θ-constacyclic code and that its dual is a punctured θ-constacyclic code. This enables us to give the general form of a parity-check matrix for module θ-codes and for module (θ, δ)-codes over Fq[X; θ, δ] where δ is a derivation over Fq. We also prove the conjecture for module θ-codes who are defined over a ring A[X; θ] where A is a finite ring. Lastly we construct self-dual θ-cyclic codes of length 2 over F4 for s ≥ 3 which are asymptotically bad and conjecture that there exists no other self-dual module θ-code of this length over F4.