The key equation for one-point codes and efficient error evaluation

In this article, the key equation and the use of error evaluator polynomials are generalized from the case of BCH codes to one-point codes. We interpret the syndrome of the error vector e as a differential ωe which has simple poles on the support of e and, in general, at the one-point Q used to define the codes. The decoding problem is to find a function f and differential φ having poles only at Q such that fωe=φ. Then if f has a simple pole at an error position P, the error value is eP=(φ/df)(P). We amend an iterative algorithm that computes a Gröbner basis for Ie, the ideal of functions vanishing on the support of e, so that it also computes the corresponding error evaluators. That is, we produce fωe for each f in the Gröbner basis.