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The question of finding an epsilon-biased set with close to optimal support size, or, equivalently, finding an explicit binary code with distance 1/2-epsilon 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 epsilon-biased set over k bits with support size O(k/epsilon^{2 o(1)}). This improves upon all previous explicit constructions which were in the order of k^2/epsilon^2, k/epsilon^3 or (k/epsilon^2)^{5/4}. The result is close to the Gilbert-Varshamov bound which is O(k/epsilon^2) and the lower bound which is $Omega(k/epsilon^2 log(1/epsilon)).
The main technical tool we use is bias amplification with the s-wide replacement product. The sum of two independent samples from a biased set is epsilon^2 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/epsilon^4). We show that amplification with a long random walk over the s-wide replacement product reduces the bias almost optimally.