Weight distributions of geometric Goppa codes

The in general hard problem of computing weight distributions of linear cod es is considered for the special class of algebraic-geometric codes, define d by Goppa in the early eighties. Known results restrict to codes from elli ptic curves. We obtain results for curves of higher genus by expressing the weight distributions in terms of L-series. The results include general pro perties of weight distributions, a method to describe and compute weight di stributions, and worked out examples for curves of genus two and three.