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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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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.

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

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.

Subject Classification

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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References

  1. Noga Alon, László Babai, and Alon Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of algorithms, 7(4):567-583, 1986. Google Scholar
  2. Noga Alon, Jeff Edmonds, and Michael Luby. Linear Time Erasure Codes with Nearly Optimal Recovery. In proceedings of the 36th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 512-519. IEEE Computer Society, 1995. Google Scholar
  3. Noga Alon and Michael Luby. A linear time erasure-resilient code with nearly optimal recovery. IEEE Transactions on Information Theory, 42(6):1732-1736, 1996. Google Scholar
  4. László Babai, Lance Fortnow, Leonid A. Levin, and Mario Szegedy. Checking Computations in Polylogarithmic Time. In Proceedings of the 23rd Annual ACM Symposium on Theory of Computing (STOC), pages 21-31. ACM Press, 1991. URL: https://doi.org/10.1145/103418.103428.
  5. Eli Ben-Sasson and Madhu Sudan. Robust locally testable codes and products of codes. Random Structures and Algorithms, 28(4):387-402, 2006. URL: https://doi.org/10.1002/rsa.20120.
  6. Eli Ben-Sasson and Michael Viderman. Tensor Products of Weakly Smooth Codes are Robust. Theory of Computing, 5(1):239-255, 2009. Google Scholar
  7. Eli Ben-Sasson and Michael Viderman. Composition of semi-LTCs by two-wise tensor products. Computational Complexity, 24(3):601-643, 2015. URL: https://doi.org/10.1007/s00037-013-0074-8.
  8. E. R. Berlekamp and L. Welch. Error correction of algebraic block codes. US Patent Number 4,633,470. Google Scholar
  9. Don Coppersmith and Atri Rudra. On the robust testability of tensor products of codes. ECCC TR05-104, 2005. URL: https://eccc.weizmann.ac.il/eccc-reports/2005/TR05-104/index.html.
  10. Irit Dinur, Madhu Sudan, and Avi Wigderson. Robust local testability of tensor products of LDPC codes. In proceedings of the 9th International Workshop on Randomization and Computation (RANDOM), pages 304-315. Springer, 2006. Google Scholar
  11. Katalin Friedl and Madhu Sudan. Some Improvements to Total Degree Tests. In proceedings of the 3rd Israel Symposium on the Theory of Computing and Systems (ISTCS), pages 190-198. IEEE Computer Society, 1995. Google Scholar
  12. Edgar N. Gilbert. A comparision of signalling alphabets. Bell System Technical Journal, 31:504-522, 1952. Google Scholar
  13. Oded Goldreich. A sample of samplers: A computational perspective on sampling. def, 1:2n, 1997. Google Scholar
  14. Oded Goldreich and Leonid A Levin. A hard-core predicate for all one-way functions. In Proceedings of the 21st annual ACM symposium on Theory of computing (STOC), pages 25-32. ACM Press, 1989. Google Scholar
  15. Oded Goldreich and Or Meir. The tensor product of two good codes is not necessarily locally testable. Information Processsing Letters, 112(8-9):351-355, 2012. Google Scholar
  16. Oded Goldreich and Madhu Sudan. Locally testable codes and PCPs of almost linear length. Journal of ACM, 53(4):558-655, 2006. Google Scholar
  17. Sivakanth Gopi, Swastik Kopparty, Rafael Oliveira, Noga Ron-Zewi, and Shubhangi Saraf. Locally Testable and Locally Correctable Codes approaching the Gilbert-Varshamov Bound. IEEE Transactions on Information Theory, 64(8):5813-5831, 2018. Google Scholar
  18. Venkatesan Guruswami and Piotr Indyk. Near-optimal linear-time codes for unique decoding and new list-decodable codes over smaller alphabets. In Proceedings of the 34th Annual ACM Symposium on Theory of Computing (STOC), pages 812-821. ACM Press, 2002. Google Scholar
  19. Venkatesan Guruswami and Piotr Indyk. Efficiently decodable codes meeting Gilbert-Varshamov bound for low rates. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithm (SODA), pages 756-757. SIAM, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.
  20. Venkatesan Guruswami and Swastik Kopparty. Explicit subspace designs. Combinatorica, 36(2):161-185, 2016. URL: https://doi.org/10.1007/s00493-014-3169-1.
  21. Venkatesan Guruswami and Atri Rudra. Explicit Codes Achieving List Decoding Capacity: Error-Correction With Optimal Redundancy. IEEE Transactions on Information Theory, 54(1):135-150, 2008. Google Scholar
  22. Venkatesan Guruswami and Chaoping Xing. List decoding Reed-Solomon, Algebraic-Geometric, and Gabidulin subcodes up to the Singleton bound. In Proceedings of the 45th annual ACM symposium on Theory of Computing (STOC), pages 843-852. ACM, 2013. Google Scholar
  23. Brett Hemenway, Noga Ron-Zewi, and Mary Wootters. Local List Recovery of High-Rate Tensor Codes & Applications. In Proceedings of the 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 2017. Google Scholar
  24. Brett Hemenway, Noga Ron-Zewi, and Mary Wootters. Local List Recovery of High-rate Tensor Codes & Applications. Electronic Colloquium on Computational Complexity (ECCC), 24:104 (revision 1), 2017. Google Scholar
  25. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In STOC '00: Proceedings of the 32nd Annual Symposium on the Theory of Computing, pages 80-86, 2000. URL: https://doi.org/10.1145/335305.335315.
  26. Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-Rate Locally Correctable and Locally Testable Codes with Sub-Polynomial Query Complexity. Journal of ACM, 64(2):11:1-11:42, 2017. URL: https://doi.org/10.1145/3051093.
  27. Swastik Kopparty, Noga Ron-Zewi, Shubhangi Saraf, and Mary Wootters. Improved List Decoding of Folded Reed-Solomon and Multiplicity Codes. In Proceedings of the 59th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE Computer Society, 2018. Google Scholar
  28. Or Meir. Combinatorial Construction of Locally Testable Codes. SIAM Journal on Computing, 39(2):491-544, 2009. Google Scholar
  29. Irving S. Reed and Gustave Solomon. Polynomial Codes over Certain Finite Fields. SIAM Journal of the Society for Industrial and Applied Mathematics, 8(2):300-304, 1960. Google Scholar
  30. Ronitt Rubinfeld and Madhu Sudan. Robust Characterization of Polynomials with Applications to Program Testing. SIAM Journal of Computing, 25(2):252-271, 1996. Google Scholar
  31. Atri Rudra and Mary Wootters. Average-radius list-recoverability of random linear codes. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 644-662. SIAM, 2018. Google Scholar
  32. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom Generators without the XOR Lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. URL: https://doi.org/10.1006/jcss.2000.1730.
  33. Christian Thommesen. The existence of binary linear concatenated codes with Reed - Solomon outer codes which asymptotically meet the Gilbert- Varshamov bound. IEEE Trans. Information Theory, 29(6):850-853, 1983. URL: https://doi.org/10.1109/TIT.1983.1056765.
  34. Paul Valiant. The tensor product of two codes is not necessarily robustly testable. In proceedings of the 9th International Workshop on Randomization and Computation (RANDOM), pages 472-481. Springer, 2005. Google Scholar
  35. R. R. Varshamov. Estimate of the number of signals in error correcting codes. Doklady Akadamii Nauk, pages 739-741, 1957. Google Scholar
  36. Michael Viderman. Strong LTCs with inverse poly-log rate and constant soundness. In proceedings of the 54th IEEE Symposium on Foundations of Computer Science (FOCS), pages 330-339. IEEE Computer Society, 2013. Google Scholar
  37. Michael Viderman. A combination of testability and decodability by tensor products. Random Structures and Algorithms, 46(3):572-598, 2015. Google Scholar
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