In [4], starting from an automorphism θ of a finite field and a skew polynomial ring , 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 where δ is a derivation over . 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 s over for s ≥ 3 which are asymptotically bad and conjecture that there exists no other self-dual module θ-code of this length over. © 2011 Springer-Verlag.
CITATION STYLE
Boucher, D., & Ulmer, F. (2011). A note on the dual codes of module skew codes. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 7089 LNCS, pp. 230–243). https://doi.org/10.1007/978-3-642-25516-8_14
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