High Rate Efficient Local List Decoding from HDX

2026-01-30

AI Summary

This work resolves several long-standing open problems in complexity theory, including nearly optimal input-preserving hardness amplification, list decoding at constant rate with depth \(O(\log N)\), and distance amplification that preserves computational complexity. To achieve this, we construct the first locally list-decodable code that simultaneously achieves high rate, high efficiency, and strong fault tolerance, based on locally spectral high-dimensional expanders (HDX). The core innovation lies in a novel multi-round belief propagation framework integrated with a strongly explicit local routing mechanism, enabling efficient local decoding in polylogarithmic time and sublogarithmic depth while maintaining seamless coordination with global error correction.

Technology Category

• Search and Optimization: Local Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• local list decoding
• high rate
• hardness amplification
• high dimensional expanders
• error tolerance

Innovation

• local list decoding
• high dimensional expanders
• belief propagation
• hardness amplification
• explicit local routing