Constructing Small-Bias Sets from Algebraic-Geometric Codes

We give an explicit construction of an epsilon-biased set over k bits of size O(k/epsilon(2) log(1/epsilon))(5/4). This improves upon previous explicit constructions when epsilon is roughly (ignoring logarithmic factors) in the range [k(-1.5), k(-0.5)]. The construction builds on an algebraic-geometric code. However, unlike previous constructions we use low-degree divisors whose degree is significantly smaller than the genus. Studying the limits of our technique, we arrive at a hypothesis that if true implies the existence of epsilon-biased sets with parameters nearly matching the lower bound, and in particular giving binary error correcting codes beating the Gilbert-Varshamov bound.