Bounding the number of points on a curve using a generalization of Weierstrass semigroups

Abstract

In this article we use techniques from coding theory to derive upper bounds for the number of rational places of the function field of an algebraic curve defined over a finite field. The used techniques yield upper bounds if the (generalized) Weierstrass semigroup for an n-tuple of places is known, even if the exact defining equation of the curve is not known. As shown in examples, this sometimes enables one to get an upper bound for the number of rational places for families of function fields. Our results extend results in [J. Pure Appl. Algebra, 213(6):1152-1156, 2009] .