Artin-Schreier curves and weights of two dimensional cyclic codes

Let GF(q) be the finite field with q elements of characteristic p, GF(q^m) be the extension of degree m>1 and f(x) be a polynomial over GF(q^m). We determine a necessary and sufficient condition for y^q-y=f(x) to have the maximum number of affine GF(qm)-rational points. Then we study the weights of 2-D cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin-Schreier curves.We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds.