Characterizing a generator polynomial matrix for the dual of a multi-twisted code

The class of multi-twisted (MT) codes generalizes the classes of cyclic, constacyclic, quasi-cyclic, quasi-twisted, and generalized quasi-cyclic codes. We establish the correspondence between MT codes over $\mathbb{F}_q$ of index $\ell$ and $\mathbb{F}_q[x]$-submodules of $(\mathbb{F}_q[x])^\ell$. Thus, a basis of an MT code exists and is used to build a generator polynomial matrix (GPM). We prove some GPM properties, for example, relationship to code dimension, the identical equation, Hermite normal form. Hence, we prove a GPM formula for the dual code of an MT code. Finally, we obtain the necessary and sufficient conditions for the self-orthogonality and self-duality of MT codes.