Improved Decoding of Expander Codes

Authors Xue Chen, Kuan Cheng, Xin Li, Minghui Ouyang

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Author Details

Xue Chen
  • University of Science and Technology of China, Anhui, China
Kuan Cheng
  • Peking University, China
Xin Li
  • Johns Hopkins University, Baltimore, MD, USA
Minghui Ouyang
  • Peking University, China

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Xue Chen, Kuan Cheng, Xin Li, and Minghui Ouyang. Improved Decoding of Expander Codes. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 43:1-43:3, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study the classical expander codes, introduced by Sipser and Spielman [M. Sipser and D. A. Spielman, 1996]. Given any constants 0 < α, ε < 1/2, and an arbitrary bipartite graph with N vertices on the left, M < N vertices on the right, and left degree D such that any left subset S of size at most α N has at least (1-ε)|S|D neighbors, we show that the corresponding linear code given by parity checks on the right has distance at least roughly {α N}/{2 ε}. This is strictly better than the best known previous result of 2(1-ε) α N [Madhu Sudan, 2000; Viderman, 2013] whenever ε < 1/2, and improves the previous result significantly when ε is small. Furthermore, we show that this distance is tight in general, thus providing a complete characterization of the distance of general expander codes. Next, we provide several efficient decoding algorithms, which vastly improve previous results in terms of the fraction of errors corrected, whenever ε < 1/4. Finally, we also give a bound on the list-decoding radius of general expander codes, which beats the classical Johnson bound in certain situations (e.g., when the graph is almost regular and the code has a high rate). Our techniques exploit novel combinatorial properties of bipartite expander graphs. In particular, we establish a new size-expansion tradeoff, which may be of independent interests.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Expander Code
  • Decoding


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