A Note on the Dual Codes of Module Skew Codes

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 2

s

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

${\mathbb F}_4$

.