Lower Bounds for 2-Query LCCs over Large Alphabet

Authors Arnab Bhattacharyya, Sivakanth Gopi, Avishay Tal

Thumbnail PDF


  • Filesize: 0.56 MB
  • 20 pages

Document Identifiers

Author Details

Arnab Bhattacharyya
Sivakanth Gopi
Avishay Tal

Cite AsGet BibTex

Arnab Bhattacharyya, Sivakanth Gopi, and Avishay Tal. Lower Bounds for 2-Query LCCs over Large Alphabet. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 30:1-30:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)


A locally correctable code (LCC) is an error correcting code that allows correction of any arbitrary coordinate of a corrupted codeword by querying only a few coordinates. We show that any 2-query locally correctable code C:{0,1}^k -> Sigma^n that can correct a constant fraction of corrupted symbols must have n >= exp(k/\log|Sigma|) under the assumption that the LCC is zero-error. We say that an LCC is zero-error if there exists a non-adaptive corrector algorithm that succeeds with probability 1 when the input is an uncorrupted codeword. All known constructions of LCCs are zero-error. Our result is tight upto constant factors in the exponent. The only previous lower bound on the length of 2-query LCCs over large alphabet was Omega((k/log|\Sigma|)^2) due to Katz and Trevisan (STOC 2000). Our bound implies that zero-error LCCs cannot yield 2-server private information retrieval (PIR) schemes with sub-polynomial communication. Since there exists a 2-server PIR scheme with sub-polynomial communication (STOC 2015) based on a zero-error 2-query locally decodable code (LDC), we also obtain a separation between LDCs and LCCs over large alphabet.
  • Locally correctable code
  • Private information retrieval
  • Szemerédi regularity lemma


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


  1. Noga Alon and Asaf Shapira. Testing subgraphs in directed graphs. Journal of Computer and System Sciences, 3(69):354-382, 2004. Google Scholar
  2. Boaz Barak, Zeev Dvir, Amir Yehudayoff, and Avi Wigderson. Rank bounds for design matrices with applications to combinatorial geometry and locally correctable codes. In Proceedings of the forty-third annual ACM symposium on Theory of computing, pages 519-528. ACM, 2011. Google Scholar
  3. 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
  4. Arnab Bhattacharyya, Zeev Dvir, Shubhangi Saraf, and Amir Shpilka. Tight lower bounds for linear 2-query LCCs over finite fields. Combinatorica, 36(1):1-36, 2016. Google Scholar
  5. Arnab Bhattacharyya and Sivakanth Gopi. Lower bounds for 2-query LCCs over large alphabet. CoRR, abs/1611.06980v1, 2016. URL: http://arxiv.org/abs/1611.06980v1.
  6. Arnab Bhattacharyya and Sivakanth Gopi. Lower bounds for constant query affine-invariant LCCs and LTCs. In 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, pages 12:1-12:17, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.12.
  7. Manuel Blum and Sampath Kannan. Designing programs that check their work. J. ACM, 42(1):269-291, 1995. Google Scholar
  8. Benny Chor, Eyal Kushilevitz, Oded Goldreich, and Madhu Sudan. Private information retrieval. J. ACM, 45(6):965-981, 1998. Google Scholar
  9. Amin Coja-Oghlan, Mikael Onsjö, and Osamu Watanabe. Propagation connectivity of random hypergraphs. The Electronic Journal of Combinatorics, 19(1):P17, 2012. Google Scholar
  10. Thomas M. Cover and Joy A. Thomas. Elements of information theory. John Wiley &Sons, 2012. Google Scholar
  11. Richard M. Dudley. Central limit theorems for empirical measures. The Annals of Probability, pages 899-929, 1978. 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. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Breaking the quadratic barrier for 3-LCC’s over the reals. In Proceedings of the 46th Annual ACM Symposium on Theory of Computing, pages 784-793. ACM, 2014. Google Scholar
  14. Zeev Dvir, Shubhangi Saraf, and Avi Wigderson. Improved rank bounds for design matrices and a new proof of Kelly’s theorem. In Forum of Mathematics, Sigma, volume 2, page e4. Cambridge Univ Press, 2014. Google Scholar
  15. Zeev Dvir and Amir Shpilka. Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM Journal on Computing, 36(5):1404-1434, 2007. Google Scholar
  16. Oded Goldreich, Howard Karloff, Leonard J. Schulman, and Luca Trevisan. Lower bounds for linear locally decodable codes and private information retrieval. Computational Complexity, 15(3):263-296, 2006. Google Scholar
  17. Rahul Jain. Towards a classical proof of exponential lower bound for 2-probe smooth codes. arXiv:cs/0607042, 2006. Google Scholar
  18. 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. ACM, 2000. Google Scholar
  19. Neeraj Kayal and Shubhangi Saraf. Blackbox polynomial identity testing for depth 3 circuits. In Foundations of Computer Science, 2009. FOCS'09. 50th Annual IEEE Symposium on, pages 198-207. IEEE, 2009. Google Scholar
  20. Iordanis Kerenidis and Ronald De Wolf. Exponential lower bound for 2-query locally decodable codes via a quantum argument. In Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, pages 106-115. ACM, 2003. Google Scholar
  21. Michel Ledoux and Michel Talagrand. Probability in Banach Spaces: isoperimetry and processes. Springer Science &Business Media, 2013. Google Scholar
  22. Richard J. Lipton. Efficient checking of computations. In Annual Symposium on Theoretical Aspects of Computer Science, pages 207-215. Springer, 1990. Google Scholar
  23. Michael Mitzenmacher and Justin Thaler. Peeling arguments and double hashing. In Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, pages 1118-1125. IEEE, 2012. Google Scholar
  24. Ryuhei Mori and Osamu Watanabe. Peeling algorithm on random hypergraphs with superlinear number of hyperedges. arXiv preprint arXiv:1506.00718, 2015. Google Scholar
  25. 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
  26. Endre Szemerédi. Regular partitions of graphs. In J. C. Bremond, J. C. Fournier, M. Las Vergnas, and D. Sotteau, editors, Proc. Colloque Internationaux CNRS 260 - Problèmes Combinatoires et Théorie des Graphes, pages 399-401, 1978. Google Scholar
  27. Amelia Taylor. The regularity method for graphs and digraphs. arXiv preprint arXiv:1406.6531, 2014. Google Scholar
  28. Stephanie Wehner and Ronald De Wolf. Improved lower bounds for locally decodable codes and private information retrieval. In International Colloquium on Automata, Languages, and Programming, pages 1424-1436. Springer, 2005. Google Scholar
  29. Sergey Yekhanin. Locally decodable codes. In Computer Science - Theory and Applications, pages 289-290. Springer, 2011. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail