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

Let F"q be a finite field. @r and @q are two automorphisms of F"q. A ring structure on the set F"q[x,y;@r,@q]={@[email protected]?a"i"jx^iy^j|a"i"[email protected]?F"q} 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 F"q[x,y;@r,@q]-submodule of the left F"q[x,y;@r,@q]-module F"q[x,y;@r,@q]/"l, where "l is the left ideal generated by x^s-1 and y^l-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.