A Uniform Upper Bound on the Length of MDS Codes from Genus-Two Curves

We study the maximal length of MDS algebraic geometry codes from genus-two curves over finite fields. Let C_ L(D,G) be such a code over Fq, with length n and dimension k. Assume that 3≤ k≤ n-3. We prove that n<(2q+10√ 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(√ q) for the maximal length. The explicit bound is strictly smaller than $q+1$ when q≥125.