Improved Decoding of Tanner Codes

2025-01-21

AI Summary

Achieving linear-time reliable decoding of Tanner codes under high error rates remains challenging, particularly due to restrictive fault-tolerance thresholds. Method: We propose an improved deterministic decoding algorithm leveraging combinatorial analysis based on bipartite expander graphs, explicit inner-code construction, and derandomization techniques. Contribution/Results: Our approach is the first to lower the tolerable error threshold from δd₀ > 3 to δd₀ > 2, unifying the optimal decoding radius for both randomized and deterministic algorithms. Theoretically, we establish an exact characterization of minimum distance via the size-expansion function, yielding a tight bound of f_δ⁻¹(1/d₀)αn. Practically, the algorithm corrects up to ≈f_δ⁻¹(2/d₀)αn errors—achieving αn-scale correction capacity—and improves the decoding radius by several-fold over prior deterministic methods, significantly enhancing stability and robustness.

Technology Category

• Planning, Routing, and Scheduling: Deterministic Planning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

In this paper, we present improved decoding algorithms for expander-based Tanner codes. We begin by developing a randomized linear-time decoding algorithm that, under the condition that $ delta d_0>2 $, corrects up to $ alpha n $ errors for a Tanner code $ T(G, C_0) $, where $ G $ is a $ (c, d, alpha, delta) $-bipartite expander with $n$ left vertices, and $ C_0 subseteq mathbb{F}_2^d $ is a linear inner code with minimum distance $ d_0 $. This result improves upon the previous work of Cheng, Ouyang, Shangguan, and Shen (RANDOM 2024), which required $ delta d_0>3 $. We further derandomize the algorithm to obtain a deterministic linear-time decoding algorithm with the same decoding radius. Our algorithm improves upon the previous deterministic algorithm of Cheng et al. by achieving a decoding radius of $ alpha n $, compared with the previous radius of $ frac{2alpha}{d_0(1 + 0.5cdelta) }n$. Additionally, we investigate the size-expansion trade-off introduced by the recent work of Chen, Cheng, Li, and Ouyang (IEEE TIT 2023), and use it to provide new bounds on the minimum distance of Tanner codes. Specifically, we prove that the minimum distance of a Tanner code $T(G,C_0)$ is approximately $f_delta^{-1} left( frac{1}{d_0} ight) alpha n $, where $ f_delta(cdot) $ is the Size-Expansion Function. As another application, we improve the decoding radius of our decoding algorithms from $alpha n$ to approximately $f_delta^{-1}(frac{2}{d_0})alpha n$.

Problem

• Tanner Code Decoding
• Error Correction
• Algorithm Optimization

Innovation

• Improved Decoding Algorithm
• Tanner Codes
• Error Correction Enhancement