2-D skew-cyclic codes over Fq[x,y;ρ,θ]Fq[x,y;ρ,θ]

Let FqFq be a finite field. ρ and θ   are two automorphisms of FqFq. A ring structure on the set Fq[x,y;ρ,θ]={∑∑aijxiyj|aij∈Fq}Fq[x,y;ρ,θ]={∑∑aijxiyj|aij∈Fq} is considered. As a generalization of 2-D cyclic codes, we propose 2-D skew-cyclic codes and prove that a 2-D skew-cyclic code is equivalent to a left Fq[x,y;ρ,θ]Fq[x,y;ρ,θ]-submodule of the left Fq[x,y;ρ,θ]Fq[x,y;ρ,θ]-module Fq[x,y;ρ,θ]/〈xs−1,yl−1〉lFq[x,y;ρ,θ]/〈xs−1,yl−1〉l, where 〈xs−1,yl−1〉l〈xs−1,yl−1〉l is the left ideal generated by xs−1xs−1 and yl−1yl−1. We introduce consistent systems in the bivariate skew polynomial ring and present some applications on 2-D skew-cyclic codes. In addition, relationships between 2-D skew-cyclic codes and 2-D cyclic codes and skew-cyclic codes are presented.