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New Near-Linear Time Decodable Codes Closer to the GV Bound

Authors Guy Blanc, Dean Doron

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Author Details

Guy Blanc
  • Computer Science Department, Stanford University, CA, USA
Dean Doron
  • Department of Computer Science, Ben Gurion University of the Negev, Beer Sheva, Israel

Funding

  • Blanc, Guy: Supported by NSF CAREER Award 1942123.
  • Doron, Dean: Part of this work was done while at Stanford, supported by NSF Award CCF-1763311.

Acknowledgements

We are grateful to Victor Lecomte and Omer Reingold for stimulating discussions and collaboration in the early stages of the project. We also thank Ori Parzanchevski, Madhur Tulsiani, and Mary Wootters and for interesting and useful discussions.

Cite AsGet BibTex

Guy Blanc and Dean Doron. New Near-Linear Time Decodable Codes Closer to the GV Bound. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 10:1-10:40, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.CCC.2022.10

Abstract

We construct a family of binary codes of relative distance 1/2-ε and rate ε² ⋅ 2^(-log^α (1/ε)) for α ≈ 1/2 that are decodable, probabilistically, in near-linear time. This improves upon the rate of the state-of-the-art near-linear time decoding near the GV bound due to Jeronimo, Srivastava, and Tulsiani, who gave a randomized decoding of Ta-Shma codes with α ≈ 5/6 [Ta-Shma, 2017; Jeronimo et al., 2021]. Each code in our family can be constructed in probabilistic polynomial time, or deterministic polynomial time given sufficiently good explicit 3-uniform hypergraphs. Our construction is based on a new graph-based bias amplification method. While previous works start with some base code of relative distance 1/2-ε₀ for ε₀ ≫ ε and amplify the distance to 1/2-ε by walking on an expander, or on a carefully tailored product of expanders, we walk over very sparse, highly mixing, hypergraphs. Study of such hypergraphs further offers an avenue toward achieving rate Ω̃(ε²). For our unique- and list-decoding algorithms, we employ the framework developed in [Jeronimo et al., 2021].

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Unique decoding
  • list decoding
  • the Gilbert-Varshamov bound
  • small-bias sample spaces
  • hypergraphs
  • expander walks

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