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

2025-04-29

AI Summary

This work studies capacity-approaching list decoding and list recovery of explicitly constructible expander codes—including Alon–Edmonds–Luby distance-amplifying codes and Sipser–Spielman Tanner codes—in nearly linear time. Addressing the longstanding challenge that algebraic techniques struggle to simultaneously achieve explicitness and efficiency, we introduce, for the first time, a systematic application of the graph regularity lemma to expander code analysis, establishing a “local base-code property → global code rigidity” lifting framework. Our approach integrates graph regularity partitioning, near-linear-time sparse graph computation, combinatorial rigidity analysis, and distance amplification. We achieve: (1) list decoding up to radius $1- ho-varepsilon$ with list size $O(1/varepsilon)$ and runtime $ ilde{O}(n)$; and (2) list decoding and list recovery up to radius $delta-varepsilon$ with constant list size. Both results attain the information-theoretic capacity limits.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Security and Privacy: Large-scale security measurements

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 $ ho$, can be list decoded to a radius $1 - ho - epsilon$ 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 $ ho$, can also be list recovered to radius $1 - ho - epsilon$ 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.

Problem

• List decoding expander-based codes up to capacity
• Near-linear time decoding for spectral expander codes
• Combinatorial bounds via graph regularity lemmas

Innovation

• Uses graph regularity lemmas for decoding
• Employs near-linear time sparse graph algorithms
• Lifts local code properties via expanders