New Near-Linear Time Decodable Codes Closer to the GV Bound

We construct a family of binary codes of relative distance [EQUATION] and rate ε 2 · 2 −log α(1/ε) for α ≈ [EQUATION] that are decodable, probabilistically, in near-linear time. This improves upon the rate of the state-of-the-art near-linear time decoding near the GV bound due to Jeronimo, Srivastava, and Tulsiani, who gave a randomized decoding of Ta-Shma codes with [EQUATION] [34, 20]. Each code in our family can be constructed in probabilistic polynomial time, or deterministic polynomial time given sufficiently good explicit 3-uniform hypergraphs. Our construction is based on a new graph-based bias amplification method. While previous works start with some base code of relative distance [EQUATION] for ε 0 > ε and amplify the distance to [EQUATION] by walking on an expander, or on a carefully tailored product of expanders, we walk over very sparse, highly mixing, hypergraphs. Study of such hypergraphs further offers an avenue toward achieving rate [EQUATION]. For our unique- and list-decoding algorithms, we employ the framework developed in [20].