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.
We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let $C_\mathcal{ L}(D,G)$ be such a code over $\mathbb F_q$, with length $n$ and dimension $k$. Assume that $3\leq k\leq n-3$. We prove that $n<(2q+10\sqrt q+16)/3$. This bound holds for every prime power $q$ and for both parities of $k$. It gives a uniform upper bound of $2q/3+O(\sqrt q)$ for the maximal length. The explicit bound is strictly smaller than $q+1$ when $q\geq125$.