Length Bounds for MDS Algebraic Geometry Codes of Arbitrary Positive Genus

We study the length of maximum distance separable algebraic geometry codes from curves of arbitrary positive genus. For fixed genus, we prove that the asymptotic ratio of the length to the field size is at most $1/2$ when both the dimension and the codimension grow faster than the logarithm of the field size. We also give explicit bounds under logarithmic dimension conditions. The bounds hold in every characteristic and are uniform in the curve and the evaluation set. The proof combines character sums on the Jacobian with an estimate for sums of distinct points.