Explicit, almost optimal, epsilon-balanced codes

The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance 1-ϵ/2 and rate close to the Gilbert-Varshamov bound, attracted a lot of attention in recent decades. In this paper we solve the problem almost optimally and show an explicit ϵ-biased set over k bits with support size O(k/ϵ2+o(1)). This improves upon all previous explicit constructions which were in the order of k2/ϵ2, k/ϵ3 or k5/4/ϵ5/2. The result is close to the Gilbert-Varshamov bound which is O(k/ϵ2) and the lower bound which is Ω(k/ϵ2 log1/ϵ). The main technical tool we use is bias amplification with the s-wide replacement product. The sum of two independent samples from an ϵ-biased set is ϵ2 biased. Rozenman and Wigderson showed how to amplify the bias more economically by choosing two samples with an expander. Based on that they suggested a recursive construction that achieves sample size O(k/ϵ4). We show that amplification with a long random walk over the s-wide replacement product reduces the bias almost optimally.