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

We construct a family of binary codes of relative distance 1/2 − ε and rate ε² · 2^{− logα(1/ε)} for α ≈ 1/2 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 α ≈ 5/6 [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 1/2 − ε₀ for ε₀ ≫ ε and amplify the distance to 1/2 − ε 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 Ω(ε²). For our unique- and list-decoding algorithms, we employ the framework developed in [20].