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When Can an Expander Code Correct Ω(n) Errors in O(n) Time?

Authors Kuan Cheng , Minghui Ouyang , Chong Shangguan , Yuanting Shen

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Author Details

Kuan Cheng
  • Center on Frontiers of Computing Studies, School of Computer Science, Peking University, China
Minghui Ouyang
  • School of Mathematical Sciences, Peking University, China
Chong Shangguan
  • Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China
  • Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, China
Yuanting Shen
  • Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, China

Funding

The research of Chong Shangguan is supported by the National Key Research and Development Program of China under Grant No. 2021YFA1001000, the National Natural Science Foundation of China under Grant Nos. 12101364 and 12231014, and the Natural Science Foundation of Shandong Province under Grant No. ZR2021QA005.

Acknowledgements

M. Ouyang would like to thank Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS, Daejeon, South Korea) for hosting his visit at the end of 2023.

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Kuan Cheng, Minghui Ouyang, Chong Shangguan, and Yuanting Shen. When Can an Expander Code Correct Ω(n) Errors in O(n) Time?. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 61:1-61:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.61

Abstract

Tanner codes are graph-based linear codes whose parity-check matrices can be characterized by a bipartite graph G together with a linear inner code C₀. Expander codes are Tanner codes whose defining bipartite graph G has good expansion property. This paper is motivated by the following natural and fundamental problem in decoding expander codes:
What are the sufficient and necessary conditions that δ and d₀ must satisfy, so that every bipartite expander G with vertex expansion ratio δ and every linear inner code C₀ with minimum distance d₀ together define an expander code that corrects Ω(n) errors in O(n) time?
For C₀ being the parity-check code, the landmark work of Sipser and Spielman (IEEE-TIT'96) showed that δ > 3/4 is sufficient; later Viderman (ACM-TOCT'13) improved this to δ > 2/3-Ω(1) and he also showed that δ > 1/2 is necessary. For general linear code C₀, the previously best-known result of Dowling and Gao (IEEE-TIT'18) showed that d₀ = Ω(cδ^{-2}) is sufficient, where c is the left-degree of G.
In this paper, we give a near-optimal solution to the above question for general C₀ by showing that δ d₀ > 3 is sufficient and δ d₀ > 1 is necessary, thereby also significantly improving Dowling-Gao’s result. We present two novel algorithms for decoding expander codes, where the first algorithm is deterministic, and the second one is randomized and has a larger decoding radius.

Subject Classification

ACM Subject Classification
  • Theory of computation → Pseudorandomness and derandomization
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
  • Mathematics of computing → Coding theory
  • Mathematics of computing → Combinatoric problems
Keywords
  • expander codes
  • expander graphs
  • linear-time decoding

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