| In [4], starting from an automorphism theta of a finite field F_q and a skew polynomial ring R=F_q[X;theta], module theta-codes are defined as left R-submodules of R/Rf where f in R. In [4] it is conjectured that an Euclidean self-dual module theta-code is a theta-constacyclic code and a proof is given in the special case when the order of theta 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 theta-code is a module theta-code if and only if it is a theta-constacyclic code. Furthermore, we establish that a module theta-code which is not theta-constacyclic is a shortened theta-constacyclic code and that its dual is a punctured theta-constacyclic code. This enables us to give the general form of a parity-check matrix for module theta-codes and for module (theta,delta)-codes over F_q[X;theta,delta] where delta is a derivation over F_q . We also prove the conjecture for module theta-codes who are defined over a ring A[X;theta] where A is a finite ring. Lastly we construct self-dual theta-cyclic codes of length 2^s over F_4 for s ≥ 3 which are asymptotically bad and conjecture that there exists no other self-dual module theta-code of this length over F_4. |
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