Document Open Access Logo

Outlaw Distributions and Locally Decodable Codes

Authors Jop Briët, Zeev Dvir, Sivakanth Gopi

PDF
Thumbnail PDF

File

LIPIcs.ITCS.2017.20.pdf
  • Filesize: 0.58 MB
  • 19 pages

Document Identifiers

Author Details

Jop Briët
Zeev Dvir
Sivakanth Gopi

Cite AsGet BibTex

Jop Briët, Zeev Dvir, and Sivakanth Gopi. Outlaw Distributions and Locally Decodable Codes. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 20:1-20:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.ITCS.2017.20

Abstract

Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. Despite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L_\infty norm) with a small number of samples. We coin the term 'outlaw distributions' for such distributions since they 'defy' the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently 'smooth' functions implies the existence of constant query LDCs and vice versa. We give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters.
Keywords
  • Locally Decodable Code
  • VC-dimension
  • Incidence Geometry
  • Cayley Hypergraphs

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Noga Alon and Vitali D. Milman. λ_1, isoperimetric inequalities for graphs, and superconcentrators. Journal of Combinatorial Theory. Series B, 38(1):73-88, 1985. Google Scholar
  2. Noga Alon and Yuval Roichman. Random Cayley graphs and expanders. Random Structures &Algorithms, 5, 1994. URL: http://citeseer.ist.psu.edu/alon97random.html.
  3. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 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, 45(1):70-122, 1998. Google Scholar
  5. Manuel Blum and Sampath Kannan. Designing programs that check their work. Journal of the ACM, 42(1):269-291, 1995. Google Scholar
  6. Jop Briët. On embeddings of 𝓁₁^k from locally decodable codes. arXiv preprint: arXiv:1611.06385, 2016. Google Scholar
  7. Jop Briët, Assaf Naor, and Oded Regev. Locally decodable codes and the failure of cotype for projective tensor products. Electronic Research Announcements in Mathematical Sciences (ERA-MS), 19:120-130, 2012. Google Scholar
  8. Jop Briët and Shravas Rao. Arithmetic expanders and deviation bounds for random tensors. arXiv preprint arXiv:1610.03428, 2016. Google Scholar
  9. Benny Chor, Eyal Kushilevitz, Oded Goldreich, and Madhu Sudan. Private information retrieval. Journal of the ACM, 45(6):965-981, 1998. Google Scholar
  10. Gil Cohen and Avishay Tal. Two structural results for low degree polynomials and applications. arXiv preprint arXiv:1404.0654, 2014. Google Scholar
  11. David Conlon and Yufei Zhao. Quasirandom Cayley graphs. Discrete Analysis, 6, 2017. Available at arXiv:1603.03025 [math.CO]. Google Scholar
  12. Zeev Dvir and Sivakanth Gopi. 2-Server PIR with sub-polynomial communication. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, pages 577-584. ACM, 2015. Google Scholar
  13. Klim Efremenko. 3-query locally decodable codes of subexponential length. In Proceedings of the fourty-first annual ACM symposium on Theory of computing (STOC 2009), pages 39-44, 2009. Google Scholar
  14. Oded Goldreich, Howard Karloff, Leonard J Schulman, and Luca Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. In Computational Complexity, 2002. Proceedings. 17th IEEE Annual Conference on, pages 143-151. IEEE, 2002. Google Scholar
  15. Parikshit Gopalan, Cheng Huang, Huseyin Simitci, and Sergey Yekhanin. On the locality of codeword symbols. IEEE Trans. Information Theory, 58(11):6925-6934, 2012. URL: http://dx.doi.org/10.1109/TIT.2012.2208937.
  16. Sivakanth Gopi, Swastik Kopparty, Rafael Oliveira, Noga Ron-Zewi, and Shubhangi Saraf. Locally testable and locally correctable codes approaching the gilbert-varshamov bound. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pages 2073-2091. SIAM, 2017. Google Scholar
  17. Shlomo Hoory, Nathan Linial, and Avi Wigderson. Expander graphs and their applications. Bull. Amer. Math. Soc., 43:439-561, 2006. Google Scholar
  18. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In Proceedings of the 32nd annual ACM symposium on Theory of computing (STOC 2000), pages 80-86. ACM Press, 2000. Google Scholar
  19. Tali Kaufman and Madhu Sudan. Sparse random linear codes are locally decodable and testable. In Foundations of Computer Science, 2007. FOCS'07. 48th Annual IEEE Symposium on, pages 590-600. IEEE, 2007. Google Scholar
  20. Iordanis Kerenidis and Ronald de Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. J. of Computer and System Sciences, 69:395-420, 2004. Preliminary version appeared in STOC'03. Google Scholar
  21. Yoshiharu Kohayakawa, Vojtěch Rödl, and Mathias Schacht. Discrepancy and eigenvalues of Cayley graphs. preprint, 2016. Available at arXiv:1602.02291 [math.CO]. Google Scholar
  22. Swastik Kopparty, Or Meir, Noga Ron-Zewi, and Shubhangi Saraf. High-rate locally-correctable and locally-testable codes with sub-polynomial query complexity. In Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, USA, June 18-21, 2016, pages 202-215, 2016. URL: http://dx.doi.org/10.1145/2897518.2897523.
  23. 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
  24. Shachar Mendelson and Roman Vershynin. Entropy and the combinatorial dimension. Invent. Math., 152(1):37-55, 2003. Google Scholar
  25. Claude E. Shannon. A mathematical theory of communication. Bell System Technical Journal, 27(3):379-423, 1948. URL: http://dx.doi.org/10.1002/j.1538-7305.1948.tb01338.x.
  26. Michel Talagrand. Upper and lower bounds for stochastic processes, volume 60 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Heidelberg, 2014. Modern methods and classical problems. Google Scholar
  27. Michael R. Tanner. Explicit concentrators from generalized n-gons. SIAM Journal on Algebraic Discrete Methods, 5(3):287-293, 1984. Google Scholar
  28. Terence Tao. Higher order Fourier analysis, volume 142. American Mathematical Society, 2012. Google Scholar
  29. Terence Tao. Algebraic combinatorial geometry: the polynomial method in arithmetic combinatorics, incidence combinatorics, and number theory. EMS Surv. Math. Sci., 1:1-46, 2014. Google Scholar
  30. Terence Tao and Van Vu. Additive Combinatorics. Cambridge University Press, 2006. Google Scholar
  31. David Woodruff. New lower bounds for general locally decodable codes. Electronic Colloquium on Computational Complexity (ECCC), 14(006), 2007. Google Scholar
  32. David Woodruff. A quadratic lower bound for three-query linear locally decodable codes over any field. J. Comput. Sci. Technol., 27(4):678-686, 2012. URL: http://dx.doi.org/10.1007/s11390-012-1254-8.
  33. Sergey Yekhanin. Towards 3-query locally decodable codes of subexponential length. In Proceedings of the 39th annual ACM symposium on Theory of computing (STOC 2007), pages 266-274, 2007. Google Scholar
  34. Sergey Yekhanin. Locally decodable codes. Foundations and Trends in Theoretical Computer Science, 6(3):139-255, 2012. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact