Algebraic algorithms for the construction of error correction codes on algebraic curves

Abstract

In this paper we present algebraic algorithms, that have been implemented in the IBM computer algebra system SCRATCHPAD2, which are used to compute the nonsingular model and the genus of a plane algebraic curve over a groundfield of positive or zero characteristic. Specifically, we describe an integral closure algorithm used to desingularize a plane algebraic curve. In the course of its execution the algorithm computes the terms which appear in the Hurwitz genus formula. The genus of the plane curve thus falls out as a byproduct. A modification of the same algorithm is also used to compute bases for principal divisors defined in the function field on the curve thus providing an automated method for construction of error corection codes on plane algebraic curves over finite fields, the application of interest. Concrete computational examples will be presented.