License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.CCC.2022.10
URN: urn:nbn:de:0030-drops-165726
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16572/
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Blanc, Guy ; Doron, Dean

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

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LIPIcs-CCC-2022-10.pdf (1 MB)

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].

BibTeX - Entry

@InProceedings{blanc_et_al:LIPIcs.CCC.2022.10,
  author =	{Blanc, Guy and Doron, Dean},
  title =	{{New Near-Linear Time Decodable Codes Closer to the GV Bound}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{10:1--10:40},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/16572},
  URN =		{urn:nbn:de:0030-drops-165726},
  doi =		{10.4230/LIPIcs.CCC.2022.10},
  annote =	{Keywords: Unique decoding, list decoding, the Gilbert-Varshamov bound, small-bias sample spaces, hypergraphs, expander walks}
}

Keywords: Unique decoding, list decoding, the Gilbert-Varshamov bound, small-bias sample spaces, hypergraphs, expander walks
Collection: 37th Computational Complexity Conference (CCC 2022)
Issue Date: 2022
Date of publication: 11.07.2022

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