On 2-Query Codeword Testing with Near-Perfect Completeness

  • Venkatesan Guruswami
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4288)


A codeword tester is a highly query-efficient spot checking procedure for ascertaining, with good confidence, proximity of a given string to its closest codeword. We consider the problem of binary codeword testing using only two queries. It is known that three queries suffice for non-trivial codeword testing with perfect completeness (where codewords must be accepted with probability 1). It is known that two queries are not enough for testing with perfect completeness, whereas two queries suffice if one relaxes the requirement of perfect completeness (this is akin to the polynomial-time decidability of 2SAT and the APX-hardness of Max 2SAT, respectively).

In this work, motivated by the parallel with 2-query PCPs and the approximability of near-satisfiable instances of Max 2SAT, we investigate 2-query testing with completeness close to 1, say 1–ε for ε→0. Our result is that, for codes of constant relative distance, such testers must also have soundness 1– O(ε) (and this is tight up to constant factors in the O(ε) term). This is to be contrasted with 2-query PCPs, where assuming the Unique Games Conjecture, one can have completeness 1–ε and soundness \(1-O(\sqrt{\varepsilon})\). Hence the ratio (1–s)/(1–c) can be super-constant for 2-query PCPs while it is bounded by a constant for 2-query LTCs. Our result also shows a similar limitation of 2-query PCPs of proximity, a notion introduced in [1].


Query Complexity Satisfying Assignment Testable Code Binary Alphabet Probabilistically Checkable Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and application to coding. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing (STOC), pp. 1–10 (2004)Google Scholar
  2. 2.
    Ben-Sasson, E., Goldreich, O., Sudan, M.: Bounds on 2-query codeword testing. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 216–227. Springer, Heidelberg (2003)Google Scholar
  3. 3.
    Bourgain, J.: On the distribution of the Fourier spectrum of boolean functions. Israel Journal of Mathematics 131, 269–276 (2002)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Charikar, M., Makarychev, K., Makarychev, Y.: Note on Max2SAT. Technical Report TR06-064, Electronic Colloquium on Computational Complexity (2006)Google Scholar
  5. 5.
    Dinur, I., Reingold, O.: Assignment Testers: Towards a combinatorial proof of the PCP-Theorem. In: Proceedings of 45th Annual Symposium on Foundations of Computer Science (FOCS), pp. 155–164 (2004)Google Scholar
  6. 6.
    Goldreich, O.: Short locally testable codes and proofs (Survey). ECCC Technical Report TR05-014 (2005)Google Scholar
  7. 7.
    Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almost linear length. In: Proceedings of 43rd Symposium on Foundations of Computer Science (FOCS), pp. 13–22 (2002)Google Scholar
  8. 8.
    Guruswami, V., Rudra, A.: Tolerant locally testable codes. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX 2005 and RANDOM 2005. LNCS, vol. 3624, pp. 306–317. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  9. 9.
    Khot, S.: On the power of unique 2-prover 1-round games. In: Proceedings of the 34th ACM Symposium on Theory of Computing (STOC), pp. 767–775 (May 2002)Google Scholar
  10. 10.
    Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: invariance and optimality. In: Proceedings of the 46th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 21–30 (2005)Google Scholar
  11. 11.
    Zwick, U.: Finding almost satisfying assignments. In: Proceedings of the 30th ACM Symposium on Theory of Computing (STOC), pp. 551–560 (May 1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of WashingtonSeattleUSA

Personalised recommendations