Generic interpolation polynomial for list decoding

We extend results of K. Lee and M.E. OʼSullivan by showing how to use Gröbner bases to find the interpolation polynomial for list decoding a one-point AG code C=CL(rP,D) on any curve X, where P is an Fq-rational point on X and D=P1+P2+⋯+Pn is the sum of other Fq-rational points on X. We then define the generic interpolation polynomial for list decoding such a code. The generic interpolation polynomial should specialize to the interpolation polynomial for most received strings. We give an example of a family of Reed–Solomon 1-error correcting codes for which a single error can be decoded by a very simple process involving substituting into the generic interpolation polynomial.