Advertisement

Abstract

We study the complexity of locally list-decoding binary error correcting codes with good parameters (that are polynomially related to information theoretic bounds). We show that computing majority over Θ(1/ε) bits is essentially equivalent to locally list-decoding binary codes from relative distance 1/2 − ε with list size at most poly (1/ε). That is, a local-decoder for such a code can be used to construct a circuit of roughly the same size and depth that computes majority on Θ(1/ε) bits. On the other hand, there is an explicit locally list-decodable code with these parameters that has a very efficient (in terms of circuit size and depth) local-decoder that uses majority gates of fan-in Θ(1/ε).

Using known lower bounds for computing majority by constant depth circuits, our results imply that every constant-depth decoder for such a code must have size almost exponential in 1/ε (this extends even to sub-exponential list sizes). This shows that the list-decoding radius of the constant-depth local-list-decoders of Goldwasser et al. [STOC07] is essentially optimal.

Keywords

locally-decodable codes list-decodable codes constant-depth circuits 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blinkovsky, V.M.: Bounds for codes in the case of list decoding of finite volume. Problems of Information Transmission 22(1), 7–19 (1986)Google Scholar
  2. 2.
    Blum, M., Micali, S.: How to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing 13(4), 850–864 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Chor, B., Kushilevitz, E., Goldreich, O., Sudan, M.: Private information retrieval. Journal of the ACM 45(6), 965–981 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Deshpande, A., Jain, R., Kavitha, T., Radhakrishnan, J., Lokam, S.V.: Better Lower Bounds for Locally Decodable Codes. In: Proceedings of the IEEE Conference on Computational Complexity, pp. 184–193 (2002)Google Scholar
  5. 5.
    Goldwasser, S., Gutfreund, D., Healy, A., Kaufman, T., Rothblum, G.N.: Verifying and decoding in constant depth. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 440–449 (2007)Google Scholar
  6. 6.
    Goldreich, O., Karloff, H.J., Schulman, L.J., Trevisan, L.: Lower bounds for linear locally decodable codes and private information retrieval. Computational Complexity 15(3), 263–296 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Goldreich, O., Levin, L.A.: A hard-core predicate for all one-way functions. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, pp. 25–32 (1989)Google Scholar
  8. 8.
    Gutfreund, D., Rothblum, G.N.: The complexity of local list decoding. Technical Report TR08-034, Electronic Colloquium on Computational Complexity (2008)Google Scholar
  9. 9.
    Guruswami, V., Vadhan, S.: A lower bound on list size for list decoding. In: Chekuri, C., Jansen, K., Rolim, J., Trevisan, L. (eds.) RANDOM 2005. LNCS, vol. 3624, pp. 318–329. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Impagliazzo, R., Jaiswal, R., Kabanets, V., Wigderson, A.: Uniform direct-product theorems: Simplified, optimized, and derandomized. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 579–588 (2008)Google Scholar
  11. 11.
    Kerenidis, I., de Wolf, R.: Exponential lower bound for 2-query locally decodable codes via a quantum argument. Journal of Computer and System Sciences 69(3), 395–420 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Katz, J., Trevisan, L.: On the efficiency of local decoding procedures for error-correcting codes. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, pp. 80–86 (2000)Google Scholar
  13. 13.
    Nisan, N., Wigderson, A.: Hardness vs. randomness. Journal of Computer and System Sciences 49, 149–167 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Obata, K.: Optimal Lower Bounds for 2-Query Locally Decodable Linear Codes. In: Proceedings of the 5th International Workshop on Randomization and Computation (RANDOM), pp. 39–50 (2002)Google Scholar
  15. 15.
    Razborov, A.A.: Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akademiya Nauk SSSR. Matematicheskie Zametki 41(4), 598–607, 623 (1987)MathSciNetGoogle Scholar
  16. 16.
    Smolensky, R.: Algebraic methods in the theory of lower bounds for boolean circuit complexity. In: Proceedings of the 19th Annual ACM Symposium on Theory of Computing, pp. 77–82 (1987)Google Scholar
  17. 17.
    Sudan, M., Trevisan, L., Vadhan, S.: Pseudorandom generators without the XOR Lemma. Journal of Computer and System Sciences 62(2), 236–266 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Shaltiel, R., Viola, E.: Hardness amplification proofs require majority. In: Proceedings of the 40th Annual ACM Symposium on Theory of Computing, pp. 589–598 (2008)Google Scholar
  19. 19.
    Trevisan, L.: List-decoding using the XOR lemma. In: Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science, pp. 126–135 (2003)Google Scholar
  20. 20.
    Trevisan, L., Vadhan, S.: Pseudorandomness and average-case complexity via uniform reductions. Computational Complexity 16(4), 361–364 (2007)CrossRefMathSciNetGoogle Scholar
  21. 21.
    Viola, E.: The complexity of constructing pseudorandom generators from hard functions. Computational Complexity 13(3-4), 147–188 (2005)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Viola, E.: The complexity of hardness amplification and derandomization. PhD thesis, Harvard University (2006)Google Scholar
  23. 23.
    Wehner, S., de Wolf, R.: Improved Lower Bounds for Locally Decodable Codes and Private Information Retrieval. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1424–1436. Springer, Heidelberg (2005)Google Scholar
  24. 24.
    Yao, A.C.: Theory and applications of trapdoor functions. In: Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pp. 80–91 (1982)Google Scholar
  25. 25.
    Yekhanin, S.: Towards 3-query locally decodable codes of subexponential length. In: Proceedings of the 39th Annual ACM Symposium on Theory of Computing, pp. 266–274 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Dan Gutfreund
    • 1
  • Guy N. Rothblum
    • 2
  1. 1.Department of Mathematics and CSAIL, MIT  
  2. 2.CSAIL, MIT  

Personalised recommendations