Explicit Almost-Optimal $\varepsilon$-Balanced Codes via Free Expander Walks

2026-01-18

AI Summary

This work investigates the explicit construction of error-correcting codes that approach the Gilbert–Varshamov (GV) bound in the regime of low rate and high relative distance. To this end, we introduce a novel method based on walks over freely extended expander graphs, where each step of the walk employs a distinct near-X-Ramanujan expander graph, thereby replacing the traditionally intricate product constructions with a significantly simpler framework. Our approach yields explicit codes achieving rate $\Omega(\varepsilon^{2+o(1)})$ at relative distance $(1-\varepsilon)/2$, whose performance approaches the GV bound within a factor of $\varepsilon^{o(1)}$. The construction not only matches the GV bound asymptotically up to sub-polynomial factors but also offers notable theoretical simplicity and full explicitness.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

We study the problem of constructing explicit codes whose rate and distance match the Gilbert-Varshamov bound in the low-rate, high-distance regime. In 2017, Ta-Shma gave an explicit family of codes where every pair of codewords has relative distance $\frac{1-\varepsilon}{2}$, with rate $\Omega(\varepsilon^{2+o(1)})$, matching the Gilbert-Varshamov bound up to a factor of $\varepsilon^{o(1)}$. Ta-Shma's construction was based on starting with a good code and amplifying its bias with walks arising from the $s$-wide-replacement product. In this work, we give an arguably simpler almost-optimal construction, based on what we call free expander walks: ordinary expander walks where each step is taken on a distinct expander from a carefully chosen sequence. This sequence of expanders is derived from the construction of near-$X$-Ramanujan graphs due to O'Donnell and Wu.

Problem

• explicit codes
• Gilbert-Varshamov bound
• low-rate high-distance regime
• relative distance
• code rate

Innovation

• free expander walks
• explicit codes
• Gilbert-Varshamov bound
• expander graphs
• almost-optimal coding