Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum für Informatik GmbH scholarly article en Kopparty, Swastik; Resch, Nicolas; Ron-Zewi, Noga; Saraf, Shubhangi; Silas, Shashwat https://www.dagstuhl.de/lipics License: Creative Commons Attribution 3.0 Unported license (CC-BY 3.0)
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URN: urn:nbn:de:0030-drops-112832
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On List Recovery of High-Rate Tensor Codes

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Abstract

We continue the study of list recovery properties of high-rate tensor codes, initiated by Hemenway, Ron-Zewi, and Wootters (FOCS'17). In that work it was shown that the tensor product of an efficient (poly-time) high-rate globally list recoverable code is approximately locally list recoverable, as well as globally list recoverable in probabilistic near-linear time. This was used in turn to give the first capacity-achieving list decodable codes with (1) local list decoding algorithms, and with (2) probabilistic near-linear time global list decoding algorithms. This also yielded constant-rate codes approaching the Gilbert-Varshamov bound with probabilistic near-linear time global unique decoding algorithms.
In the current work we obtain the following results:
1) The tensor product of an efficient (poly-time) high-rate globally list recoverable code is globally list recoverable in deterministic near-linear time. This yields in turn the first capacity-achieving list decodable codes with deterministic near-linear time global list decoding algorithms. It also gives constant-rate codes approaching the Gilbert-Varshamov bound with deterministic near-linear time global unique decoding algorithms.
2) If the base code is additionally locally correctable, then the tensor product is (genuinely) locally list recoverable. This yields in turn (non-explicit) constant-rate codes approaching the Gilbert-Varshamov bound that are locally correctable with query complexity and running time N^{o(1)}. This improves over prior work by Gopi et. al. (SODA'17; IEEE Transactions on Information Theory'18) that only gave query complexity N^{epsilon} with rate that is exponentially small in 1/epsilon.
3) A nearly-tight combinatorial lower bound on output list size for list recovering high-rate tensor codes. This bound implies in turn a nearly-tight lower bound of N^{Omega(1/log log N)} on the product of query complexity and output list size for locally list recovering high-rate tensor codes.

BibTeX - Entry

@InProceedings{kopparty_et_al:LIPIcs:2019:11283,
  author =	{Swastik Kopparty and Nicolas Resch and Noga Ron-Zewi and Shubhangi Saraf and Shashwat Silas},
  title =	{{On List Recovery of High-Rate Tensor Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{68:1--68:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Dimitris Achlioptas and L{\'a}szl{\'o} A. V{\'e}gh},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/11283},
  URN =		{urn:nbn:de:0030-drops-112832},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.68},
  annote =	{Keywords: Coding theory, Tensor codes, List-decoding and recovery, Local codes}
}

Keywords: Coding theory, Tensor codes, List-decoding and recovery, Local codes
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)
Issue date: 2019
Date of publication: 17.09.2019


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