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−є/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 є-biased set over k bits with support size O(k/є2+o(1)). This improves upon all previous explicit constructions which were in the order of k2/є2, k/є3 or k5/4/є5/2. The result is close to the Gilbert-Varshamov bound which is O(k/є2) and the lower bound which is Ω(k/є2 log1/є). The main technical tool we use is bias amplification with the s-wide replacement product. The sum of two independent samples from an є-biased set is є2 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/є4). We show that amplification with a long random walk over the s-wide replacement product reduces the bias almost optimally.