Generator matrix for two-dimensional cyclic codes of arbitrary length

Two-dimensional cyclic codes of length $n=\ell s$ over the finite field $\mathbb{F}$ are ideals of the polynomial ring $\frac{\mathbb{F}[x,y]}{< x^{s}-1,y^{\ell}-1 >}$. The aim of this paper, is to present a novel method to study the algebraic structure of two-dimensional cyclic codes of any length $n=\ell s$ over the finite field $\mathbb{F}$. By using this method, we find the generator polynomials for ideals of $\frac{\mathbb{F}[x,y]}{< x^{s}-1,y^{\ell}-1 >}$ corresponding to two dimensional cyclic codes. These polynomials will be applied to obtain the generator matrix for two- dimensional cyclic codes.