An error-correcting code is said to be locally testable if it has an efficient spot-checking procedure that can distinguish codewords from strings that are far from every codeword, looking at very few locations of the input in doing so. Locally testable codes (LTCs) have generated a lot of interest over the years, in large part due to their connection to Probabilistically checkable proofs (PCPs). The ability to correct errors that occur during transmission is one of the big advantages of using a code. Hence, from a coding-theoretic angle, local testing is potentially more useful if in addition to accepting codewords, it also accepts strings that are close to a codeword (in contrast, local testers can have arbitrary behavior on such strings, which potentially annuls the benefits of error-correction). This would imply that when the tester accepts, one can follow-up the testing with a (more expensive) decoding procedure to correct the errors and recover the transmitted codeword, while if the tester rejects, we can save the effort of running the more expensive decoding algorithm.

In this work, we define such testers, which we call tolerant testers following some recent work in property testing [13]. We revisit some recent constructions of LTCs and show how one can make them locally testable in a tolerant sense. While we do not optimize the parameters, the main message from our work is that there are explicit tolerant LTCs with similar parameters to LTCs.


Tolerant Tester Block Length Testable Code Boolean Circuit Oracle Access 
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.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the intractibility of approximation problems. Journal of the ACM 45(3), 501–555 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Babai, L., Shpilka, A., Stefankovic, D.: Locally testable cyclic codes. In: Proceedings of 44th Annual Symposium on Foundations of Computer Science (FOCS), pp. 116–125 (2003)Google Scholar
  4. 4.
    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
  5. 5.
    Ben-Sasson, E., Sudan, M.: Robust locally testable codes and products of codes. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 286–297. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Ben-Sasson, E., Sudan, M.: Simple PCPs with poly-log rate and query complexity. In: Proceedings of 37th ACM Symposium on Theory of Computing (STOC), pp. 266–275 (2005)Google Scholar
  7. 7.
    Dinur, I.: The PCP theorem by gap amplification. In ECCC Technical Report TR05-046 (2005)Google Scholar
  8. 8.
    Fischer, E., Fortnow, L.: Tolerant versus intolerant testing for boolean properties. In: Proceedings of the 20th IEEE Conference on Computational Complexity (2005) (to appear) Google Scholar
  9. 9.
    Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proceedings of the 3rd Israel Symp. on Theory and Computing Systems (ISTCS), pp. 190–198 (1995)Google Scholar
  10. 10.
    Goldreich, O.: Short locally testable codes and proofs (Survey). ECCC Technical Report TR05-014 (2005)Google Scholar
  11. 11.
    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
  12. 12.
    Kaufman, T., Ron, D.: Testing polynomials over general fields. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 413–422 (2004)Google Scholar
  13. 13.
    Parnas, M., Ron, D., Rubinfeld, R.: Tolerant property testing and distance approximation. In ECCC Technical Report TR04-010 (2004)Google Scholar
  14. 14.
    Polishchuk, A., Spielman, D.A.: Nearly-linear size holographic proofs. In: Proceedings of the 26th Annual ACM Symposium on Theory of Computing (STOC), pp. 194–203 (1994)Google Scholar
  15. 15.
    Sipser, M., Spielman, D.: Expander codes. IEEE Transactions on Information Theory 42(6), 1710–1722 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Sudan, M.: Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems. LNCS, vol. 1001. Springer, Heidelberg (1996)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Venkatesan Guruswami
    • 1
  • Atri Rudra
    • 1
  1. 1.Department of Computer Science & EngineeringUniversity of WashingtonSeattleUSA

Personalised recommendations