List Decoding Expander-Based Codes up to Capacity in Near-Linear Time

Abstract: We give a new framework based on graph regularity lemmas, for list decoding and list recovery of codes based on spectral expanders. Using existing algorithms for computing regularity decompositions of sparse graphs in (randomized) near-linear time, and appropriate choices for the constant-sized inner/base codes, we prove the following: - Expander-based codes constructed using the distance amplification technique of Alon, Edmonds and Luby [FOCS 1995] with rate $\rho$, can be list decoded to a radius $1 - \rho - \epsilon$ in near-linear time. By known results, the output list has size $O(1/\epsilon)$. - The above codes of Alon, Edmonds and Luby, with rate $\rho$, can also be list recovered to radius $1 - \rho - \epsilon$ in near-linear time, with constant-sized output lists. - The Tanner code construction of Sipser and Spielman [IEEE Trans. Inf. Theory 1996] with distance $\delta$, can be list decoded to radius $\delta - \epsilon$ in near-linear time, with constant-sized output lists. Our results imply novel combinatorial as well as algorithmic bounds for each of the above explicit constructions. All of these bounds are obtained via combinatorial rigidity phenomena, proved using (weak) graph regularity. The regularity framework allows us to lift the list decoding and list recovery properties for the local base codes, to the global codes obtained via the above constructions.