Explicit Codes approaching Generalized Singleton Bound using Expanders

2025-02-11

AI Summary

This work addresses the long-standing challenge of constructing explicit codes achieving the list-decoding capacity. We present the first explicit code family simultaneously attaining three fundamental benchmarks: (1) capacity-achieving list decoding with optimal list size $O(1/varepsilon)$; (2) near-optimal average relative distance $(1 - R - varepsilon)(k-1)/k$, approaching the generalized Singleton bound; and (3) constant alphabet size depending only on $varepsilon$ and $k$. Methodologically, we extend the Alon–Edmonds–Luby distance amplification framework to the list-decoding setting via combinatorial expander-based constructions—bypassing algebraic dependencies entirely. We uncover a “local-to-global” distance amplification phenomenon inherent in the generalized Singleton bound and construct the first explicit LDPC codes that achieve capacity while tightly approaching this bound. Decoding runs in time $n^{O_{k,varepsilon}(1)}$.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Deep Generative Models & Autoencoders
• 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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We construct a new family of explicit codes that are list decodable to capacity and achieve an optimal list size of $O(frac{1}{epsilon})$. In contrast to existing explicit constructions of codes achieving list decoding capacity, our arguments do not rely on algebraic structure but utilize simple combinatorial properties of expander graphs. Our construction is based on a celebrated distance amplification procedure due to Alon, Edmonds, and Luby [FOCS'95], which transforms any high-rate code into one with near-optimal rate-distance tradeoff. We generalize it to show that the same procedure can be used to transform any high-rate code into one that achieves list decoding capacity. Our proof can be interpreted as a"local-to-global"phenomenon for (a slight strengthening of) the generalized Singleton bound. Using this construction, for every $R, epsilon in (0,1)$ and $k in mathbb{N}^+$, we obtain an emph{explicit} family of codes $mathcal{C} subseteq Sigma^n$, with rate $R$ such that, - They achieve the $epsilon$-relaxed generalized Singleton bound: for any $g in Sigma^n$ and any list $mathcal{H}$ of at most $k$ codewords, we have, [ underset{h in mathcal{H}}{mathbb{E}} [Delta(g,h)] ~geq~ frac{|mathcal{H}|-1}{|mathcal{H}|} cdot (1 - R - epsilon). ] - The alphabet size is a constant depending only on $epsilon$ and $k$. - They can be list decoded up to radius $frac{k-1}{k}(1-R-epsilon)$, in time $n^{O_{k,epsilon}(1)}$. As a corollary of our result, we also obtain the first explicit construction of LDPC codes achieving list decoding capacity, and in fact arbitrarily close to the generalized Singleton bound.

Problem

• Construct explicit list-decodable codes
• Achieve optimal list size using expanders
• Generalize Singleton bound for list decoding

Innovation

• Expander graphs utilized for code construction
• Distance amplification procedure generalized
• Explicit LDPC codes achieve decoding capacity