Two Dimensional (α, β)-Constacyclic Codes of arbitrary length over a Finite Field.

In this paper we characterize the algebraic structure of two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes of arbitrary length $$s\ell $$ and of their duals over a finite field $$\mathbb{F}_q $$ , where $$\alpha ,\beta$$ are non zero elements of $$\mathbb{F}_q $$ . For $$\alpha ,\beta \in \{1,-1\}$$ , we give necessary and sufficient conditions for a two-dimensional $$(\alpha ,\beta )$$ -constacyclic code to be self-dual. We also show that a two-dimensional $$(\alpha ,1 )$$ -constacyclic code $${\mathcal {C}}$$ of length $$n=s\ell $$ cannot be self-dual if $$\gcd (s,q)= 1$$ . Finally, we give some examples of self-dual, isodual, MDS and quasi-twisted codes corresponding to two-dimensional $$(\alpha ,\beta )$$ -constacyclic codes.