Tolerant Locally Testable Codes
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 . 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.
KeywordsTolerant Tester Block Length Testable Code Boolean Circuit Oracle Access
Unable to display preview. Download preview PDF.
- 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.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
- 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.Dinur, I.: The PCP theorem by gap amplification. In ECCC Technical Report TR05-046 (2005)Google Scholar
- 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.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.Goldreich, O.: Short locally testable codes and proofs (Survey). ECCC Technical Report TR05-014 (2005)Google Scholar
- 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.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.Parnas, M., Ron, D., Rubinfeld, R.: Tolerant property testing and distance approximation. In ECCC Technical Report TR04-010 (2004)Google Scholar
- 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
- 16.Sudan, M.: Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems. LNCS, vol. 1001. Springer, Heidelberg (1996)Google Scholar