We continue the investigation of locally testable codes, i.e., error-correcting codes for whom membership of a given word in the code can be tested probabilistically by examining it in very few locations. We give two general results on local testability: First, motivated by the recently proposed notion of robust probabilistically checkable proofs, we introduce the notion of robust local testability of codes. We relate this notion to a product of codes introduced by Tanner, and show a very simple composition lemma for this notion. Next, we show that codes built by tensor products can be tested robustly and somewhat locally, by applying a variant of a test and proof technique introduced by Raz and Safra in the context of testing low-degree multivariate polynomials (which are a special case of tensor codes).

Combining these two results gives us a generic construction of codes of inverse polynomial rate, that are testable with poly-logarithmically many queries. We note these locally testable tensor codes can be obtained from any linear error correcting code with good distance. Previous results on local testability, albeit much stronger quantitatively, rely heavily on algebraic properties of the underlying codes.


Tensor Product Linear Code Local Testability Query Complexity Product Code 
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.
    Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing Reed-Muller codes. In: Proc. RANDOM 2003, pp. 188–199 (2003)Google Scholar
  2. 2.
    Arora, S.: Probabilistic checking of proofs and the hardness of approximation problems. PhD thesis, University of California at Berkeley (1994) Google Scholar
  3. 3.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3), 501–555 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arora, S., Sudan, M.: Improved low-degree testing and its applications. In: Proc. STOC 1997, El Paso, Texas, May 4-6, pp. 485–495 (1997)Google Scholar
  6. 6.
    Babai, L., Fortnow, L., Levin, L.A., Szegedy, M.: Checking computations in polylogarithmic time. In: Proc. STOC 1991, pp. 21–32 (1991)Google Scholar
  7. 7.
    Babai, L., Fortnow, L., Lund, C.: Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity 1(1), 3–40 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M., Sudan, M.: Linearity testing over characteristic two. IEEE Transactions on Information Theory 42(6), 1781–1795 (1996)zbMATHCrossRefGoogle Scholar
  9. 9.
    Bellare, M., Goldwasser, S., Lund, C., Russell, A.: Efficient probabilistically checkable proofs and applications to approximation. In: Proc. STOC 1993, pp. 294–304. ACM, New York (1993)Google Scholar
  10. 10.
    Bellareand, M., Sudan, M.: Improved non-approximability results. In: Proc. STOC 1994, Montreal, Quebec, Canada, May 23-25, pp. 184–193 (1994)Google Scholar
  11. 11.
    Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. In: Proc. STOC0 (2004) (to appear)Google Scholar
  12. 12.
    Ben-Sasson, E., Sudan, M.: Robust Locally Testable Codes and Products of Codes. Electronic Colloquium on Computational Complexity, Available at
  13. 13.
    Ben-Sasson, E., Sudan, M., Vadhan, S., Wigderson, A.: Randomness efficient low-degree tests and short PCPs via -biased sets. In: Proc. STOC 2003, pp. 612–621 (2003)Google Scholar
  14. 14.
    Blum, M., Luby, M., Rubinfeld, R.: Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences 47(3), 549–595 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dinur, I., Reingold, O.: Assignment-Testers: Towards a Combinatorial. Proof of the PCP-Theorem (2004) (manuscript) Google Scholar
  16. 16.
    Feige, U., Goldwasser, S., Lovasz, L., Safra, S., Szegedy, M.: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43(2), 268–292 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Friedl, K., Hatsagi, Z., Shen, A.: Low-degree tests. In: Proc. SODA 1994, pp. 57–64 (1994)Google Scholar
  18. 18.
    Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proc. Israel STCS, Tel Aviv, Israel, January 4-6, pp. 190–198 (1995)Google Scholar
  19. 19.
    Goldreich, O., Safra, M.: A Combinatorial Consistency Lemma with application to the PCP Theorem. SIAM Jour. on Comp. 29(4), 1132–1154 (1999)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almostlinear length. In: Proc. FOCS 2002, Vancouver, Canada, 16-19 November (2002)Google Scholar
  21. 21.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. Elsevier/North-Holland, Amsterdam (1981)Google Scholar
  22. 22.
    Polishchuk, A., Spielman, D.A.: Nearly linear-size holographic proofs. In: Proc. STOC 1994, Montreal, Canada, pp. 194–203 (1994)Google Scholar
  23. 23.
    Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. STOC 1997, pp. 475–484. ACM Press, New York (1997)Google Scholar
  24. 24.
    Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM J. Comp. 25(2), 252–271 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Sudan, M.: Algorithmic introduction to coding theory. Lecture notes (2001), Available from
  26. 26.
    Michael Tanner, R.: A recursive approach to low complexity codes. IEEE Transactions of Information Theory 27(5), 533–547 (1981)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Eli Ben-Sasson
    • 1
  • Madhu Sudan
    • 2
  1. 1.Radcliffe Institute for Advanced StudyCambridgeUSA
  2. 2.MIT and Radcliffe IAS, The Stata Center Rm. G640CambridgeUSA

Personalised recommendations