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Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC

Authors Gil Cohen, Tal Yankovitz

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Author Details

Gil Cohen
  • Department of Computer Science, Tel Aviv University, Israel
Tal Yankovitz
  • Department of Computer Science, Tel Aviv University, Israel

Funding

The research leading to these results has received funding from the ERC starting grant 949499, the Israel Science Foundation grant 1569/18 and from the Azrieli Faculty Fellowship.

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Gil Cohen and Tal Yankovitz. Rate Amplification and Query-Efficient Distance Amplification for Linear LCC and LDC. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 1:1-1:57, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.CCC.2021.1

Abstract

The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ρ and, say, constant distance to an asymptotically good LCC with q^poly(1/ρ) queries. Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance δ and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/δ asymptotically good LDC. This improves upon the poly(1/δ) overhead obtained by the AEL distance amplification procedure [Alon and Luby, 1996; Alon et al., 1995]. Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively.

Subject Classification

ACM Subject Classification
  • Theory of computation → Error-correcting codes
Keywords
  • Locally decodable codes
  • Locally correctable codes

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References

  1. Noga Alon, Jeff Edmonds, and Michael Luby. Linear time erasure codes with nearly optimal recovery. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 512-519. IEEE, 1995. Google Scholar
  2. 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
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM (JACM), 45(3):501-555, 1998. Google Scholar
  4. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM (JACM), 45(1):70-122, 1998. Google Scholar
  5. László Babai, Lance Fortnow, Leonid A Levin, and Mario Szegedy. Checking computations in polylogarithmic time. In Proceedings of the twenty-third annual ACM symposium on Theory of computing, pages 21-32, 1991. Google Scholar
  6. Laszlo Babai, Lance Fortnow, Noam Nisan, and Avi Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3(4):307-318, 1993. Google Scholar
  7. Omer Barkol, Yuval Ishai, and Ronny Roth. Locally decodable codes and their applications. PhD thesis, Computer Science Department, Technion, 2008. Google Scholar
  8. Donald Beaver and Joan Feigenbaum. Hiding instances in multioracle queries. In Annual Symposium on Theoretical Aspects of Computer Science, pages 37-48. Springer, 1990. Google Scholar
  9. Mihir Bellare and John Rompel. Randomness-efficient oblivious sampling. In Proceedings of the 35th Annual Symposium on Foundations of Computer Science, 1994, pages 276-287. IEEE, 1994. Google Scholar
  10. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 73-83, 1990. Google Scholar
  11. Benny Chor, Oded Goldreich, Eyal Kushilevitz, and Madhu Sudan. Private information retrieval. In Proceedings of IEEE 36th Annual Foundations of Computer Science, pages 41-50. IEEE, 1995. Google Scholar
  12. Zeev Dvir. On matrix rigidity and locally self-correctable codes. computational complexity, 20(2):367-388, 2011. Google Scholar
  13. Zeev Dvir, Parikshit Gopalan, and Sergey Yekhanin. Matching vector codes. SIAM Journal on Computing, 40(4):1154-1178, 2011. Google Scholar
  14. Klim Efremenko. 3-query locally decodable codes of subexponential length. SIAM Journal on Computing, 41(6):1694-1703, 2012. Google Scholar
  15. Peter Gemmell, Richard Lipton, Ronitt Rubinfeld, Madhu Sudan, and Avi Wigderson. Self-testing/correcting for polynomials and for approximate functions. In STOC, volume 91, pages 32-42. Citeseer, 1991. Google Scholar
  16. Peter Gemmell and Madhu Sudan. Highly resilient correctors for polynomials. Information processing letters, 43(4):169-174, 1992. Google Scholar
  17. Oded Goldreich. A sample of samplers: A computational perspective on sampling. In Studies in Complexity and Cryptography, pages 302-332. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22670-0_24.
  18. Oded Goldreich and Madhu Sudan. Locally testable codes and PCPs of almost-linear length. Journal of the ACM (JACM), 53(4):558-655, 2006. Google Scholar
  19. 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
  20. Alan Guo, Swastik Kopparty, and Madhu Sudan. New affine-invariant codes from lifting. In Proceedings of the 4th conference on Innovations in Theoretical Computer Science, pages 529-540. ACM, 2013. Google Scholar
  21. Venkatesan Guruswami, Atri Rudra, and Madhu Sudan. Essential coding theory. Draft available at http://www.cse.buffalo.edu/~atri/courses/coding-theory/book, 2012.
  22. Brett Hemenway, Rafail Ostrovsky, and Mary Wootters. Local correctability of expander codes. Information and Computation, 243:178-190, 2015. Google Scholar
  23. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the thirty-second annual ACM symposium on Theory of computing, pages 80-86, 2000. Google Scholar
  24. Tali Kaufman and Madhu Sudan. Sparse random linear codes are locally decodable and testable. In 48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07), pages 590-600. IEEE, 2007. Google Scholar
  25. Kiran S Kedlaya and Sergey Yekhanin. Locally decodable codes from nice subsets of finite fields and prime factors of mersenne numbers. SIAM Journal on Computing, 38(5):1952-1969, 2009. Google Scholar
  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 the ACM (JACM), 64(2):11, 2017. Google Scholar
  27. Swastik Kopparty, Shubhangi Saraf, and Sergey Yekhanin. High-rate codes with sublinear-time decoding. Journal of the ACM (JACM), 61(5):28, 2014. Google Scholar
  28. Ray Li and Mary Wootters. Lifted multiplicity codes and the disjoint repair group property. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  29. Richard J. Lipton. Efficient checking of computations. In Annual Symposium on Theoretical Aspects of Computer Science, pages 207-215. Springer, 1990. Google Scholar
  30. Irving S Reed. A class of multiple-error-correcting codes and the decoding scheme. Technical report, Massachusetts inst of tech Lexington Lincoln lab, 1953. Google Scholar
  31. Omer Reingold, Salil Vadhan, and Avi Wigderson. Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Electronic Colloquium on Computational Complexity (ECCC), page 18, 2001. URL: https://eccc.weizmann.ac.il/report/2001/018/.
  32. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, 1996. Google Scholar
  33. C. E. Shannon. A mathematical theory of communication. ACM SIGMOBILE Mobile Computing and Communications Review, 5(1):3-55, 2001. Originally appeared in Bell System Tech. J. 27:379-423, 623-656, 1948. Google Scholar
  34. Carl Siegel. Über die classenzahl quadratischer zahlkörper. Acta Arithmetica, 1(1):83-86, 1935. Google Scholar
  35. Madhu Sudan, Luca Trevisan, and Salil Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. Google Scholar
  36. Luca Trevisan. List-decoding using the XOR lemma. In 44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings., pages 126-135. IEEE, 2003. Google Scholar
  37. Arnold Walfisz. Zur additiven zahlentheorie. ii. Mathematische Zeitschrift, 40(1):592-607, 1936. Google Scholar
  38. David Woodruff. New lower bounds for general locally decodable codes. In Electronic Colloquium on Computational Complexity (ECCC), volume 14, 2007. Google Scholar
  39. S. Yekhanin. Locally decodable codes. In International Computer Science Symposium in Russia, pages 289-290. Springer, 2011. Google Scholar
  40. Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. Journal of the ACM (JACM), 55(1):1-16, 2008. Google Scholar
  41. Kalina Petrova Zeev Dvir. Lecture 1: Introduction. Lecture notes: https://www.cs.princeton.edu/~zdvir/LDCnotes/LDC1.pdf, year = 2016.
  42. Kalina Petrova Zeev Dvir. Lecture 4: Lower bounds for r-query LDCs. Lecture notes: https://www.cs.princeton.edu/~zdvir/LDCnotes/LDC4.pdf, 2016.
  43. Victor Vasilievich Zyablov. An estimate of the complexity of constructing binary linear cascade codes. Problemy Peredachi Informatsii, 7(1):5-13, 1971. Google Scholar
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