Artin–Schreier curves and weights of two-dimensional cyclic codes

Abstract

Let F q be the finite field with q elements of characteristic p, F q m be the extension of degree m>1 and f( x) be a polynomial over F q m . The maximum number of affine F q m -rational points that a curve of the form y q − y= f( x) can have is q m+1 . We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (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.