A Note on the Dual Codes of Module Skew Codes

Abstract

In [4], starting from an automorphism θ of a finite field ${\mathbb F}_q$ and a skew polynomial ring $R={\mathbb F}_q[X;\theta]$ , 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 ${\mathbb F}_q[X;\theta,\delta]$ where δ is a derivation over ${\mathbb F}_q$ . 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 2s over ${\mathbb F}_4$ for s ≥3 which are asymptotically bad and conjecture that there exists no other self-dual module θ -code of this length over ${\mathbb F}_4$ .