High Rate Efficient Local List Decoding from HDX

TL;DR

This paper introduces highly efficient, locally decodable list codes based on high-dimensional expanders, achieving near-optimal rate, error tolerance, and computational efficiency, with applications in complexity theory and cryptography.

Contribution

It presents the first locally list decodable codes with near-theoretic limits, utilizing a novel belief propagation framework and explicit local routing on HDX.

Findings

Codes run in polylogarithmic time and sub-logarithmic depth.

Achieves near-optimally input-preserving hardness amplification.

Resolves several longstanding problems in coding and complexity theory.

Abstract

We construct the first (locally computable, approximately) locally list decodable codes with rate, efficiency, and error tolerance approaching the information theoretic limit, a core regime of interest for the complexity theoretic task of hardness amplification. Our algorithms run in polylogarithmic time and sub-logarithmic depth, which together with classic constructions in the unique decoding (low-noise) regime leads to the resolution of several long-standing problems in coding and complexity theory: 1. Near-optimally input-preserving hardness amplification (and corresponding fast PRGs) 2. Constant rate codes with $\log(N)$-depth list decoding (RNC$^1$) 3. Complexity-preserving distance amplification Our codes are built on the powerful theory of (local-spectral) high dimensional expanders (HDX). At a technical level, we make two key contributions. First, we introduce a new framework for ($\mathrm{polylog(N)}$-round) belief propagation on HDX that leverages a mix of local correction and global expansion to control error build-up while maintaining high rate. Second, we introduce the notion of strongly explicit local routing on HDX, local algorithms that given any two target vertices, output a random path between them in only polylogarithmic time (and, preferably, sub-logarithmic depth). Constructing such schemes on certain coset HDX allows us to instantiate our otherwise combinatorial framework in polylogarithmic time and low depth, completing the result.