Two dimensional constacyclic codes of arbitrary length over finite fields

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