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On List Recovery of High-Rate Tensor Codes

Authors Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, Shashwat Silas

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ACM Subject Classification
  • Mathematics of computing → Coding theory
  • Theory of computation → Pseudorandomness and derandomization
Keywords
  • Coding theory
  • Tensor codes
  • List-decoding and recovery
  • Local 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.

Cite As Get BibTex

Swastik Kopparty, Nicolas Resch, Noga Ron-Zewi, Shubhangi Saraf, and Shashwat Silas. On List Recovery of High-Rate Tensor Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 68:1-68:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2019.68

Author Details

Swastik Kopparty
  • Department of Mathematics and Department of Computer Science, Rutgers University, NJ, USA
Nicolas Resch
  • Department of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA
Noga Ron-Zewi
  • Department of Computer Science, University of Haifa, Israel
Shubhangi Saraf
  • Department of Mathematics and Department of Computer Science, Rutgers University, NJ, USA
Shashwat Silas
  • Department of Computer Science, Stanford University, CA, USA

Funding

  • Kopparty, Swastik: Research supported in part by NSF grants CCF-1253886, CCF-1540634, CCF-1814409 and CCF-1412958, and BSF grant 2014359. Some of this research was done while visiting the Institute for Advanced Study.
  • Resch, Nicolas: Research supported in part by NSF-BSF grant CCF-1814629 and 2017732, NSERC grant CGSD2-502898, NSF grants CCF- 1422045, CCF-1527110, CCF-1618280, CCF-1814603, CCF-1910588, NSF CAREER award CCF-1750808 and a Sloan Research Fellowship.
  • Ron-Zewi, Noga: Research supported in part by NSF-BSF grant CCF-1814629 and 2017732.
  • Saraf, Shubhangi: Research supported in part by NSF grants CCF-1350572, CCF-1540634 and CCF-1412958, BSF grant 2014359, a Sloan research fellowship and the Simons Collaboration on Algorithms and Geometry. Some of this research was done while visiting the Institute for Advanced Study.
  • Silas, Shashwat: Research supported in part by NSF-BSF grant CCF-1814629 and 2017732 and a Google Fellowship in the School of Engineering at Stanford.

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