The question of whether the Gilbert-Varshamov bound (1952, 1957) gives the optimal tradeoff between rate and distance is one of the most fundamental in coding theory. It is a central instance of the broader question of how far algebraic structure can surpass the benchmark provided by the probabilistic method.

In a celebrated breakthrough, Tsfasman, Vladut, and Zink (1982) combined Goppa's construction of algebraic-geometric codes, introduced a year earlier, with deep results from algebraic geometry and number theory to beat the asymptotic Gilbert-Varshamov bound over sufficiently large fields, starting at size $49$. Building on their work, Zinoviev and Litsyn (1985) showed that nonlinear codes beat this bound over every alphabet of size at least $46$.

We prove the existence of a family of codes beating the asymptotic Gilbert-Varshamov bound over an alphabet of size $42$, improving a threshold that has stood for over forty years. Like Zinoviev and Litsyn, we restrict the allowed symbols in each coordinate of an algebraic-geometric code. We then improve the distance by removing relatively few codewords. This is made possible by averaging the number of surviving close pairs over divisor classes and using the zeta function of the underlying function field to bound this average.