Abstract
We study tolerant linearity testing under general distributions. Given groups G and H, a distribution μ on G, and oracle access to a function f:G → H, we consider the task of approximating the smallest μ-distance of f to a homomorphism h:G → H, where the μ-distance between f and h is the probability that f(x) ≠ h(x) when x is drawn according to the distribution μ. This question is intimately connected to local testability of linear codes.
In this work, we give a general sufficient condition on the distribution μ for linearity to be tolerantly testable with a constant number of queries. Using this condition we show that linearity is tolerantly testable for several natural classes of distributions including low bias, symmetric and product distributions. This gives a new and simple proof of a result of Kaufman and Sudan which shows that sparse, unbiased linear codes over \(\mathbb{Z}_2^n\) are locally testable.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing low-degree polynomials over GF(2). In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 188–199. Springer, Heidelberg (2003)
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)
Ben-Sasson, E., Goldreich, O., Harsha, P., Sudan, M., Vadhan, S.: Robust PCPs of proximity, shorter PCPs and applications to coding. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 1–10. ACM Press, New York (2004)
Ben-Sasson, E., Sudan, M.: Short PCPs with poly-log rate and query complexity. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 266–275. ACM Press, New York (2005)
Ben-Sasson, E., Sudan, M., Vadhan, S., Wigderson, A.: Randomness efficient low-degree tests and short PCPs via ε-biased sets. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing, pp. 612–621. ACM Press, New York (2003)
Dinur, I.: The PCP theorem by gap amplification. In: Proceedings of the 38th Annual ACM Symposium on Theory of Computing, pp. 241–250. ACM Press, New York (2006); Preliminary version appeared as an ECCC Technical Report TR05-046
Dinur, I., Safra, S.: The importance of being biased. In: Proceedings on 34th Annual ACM Symposium on Theory of Computing, Montreal, Quebec, Canada, May 19-21, 2002, pp. 33–42 (2002)
Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. JACM 45(4), 653–750 (1998)
Goldreich, O.: Three xor-lemmas - an exposition. Electronic Colloquium on Computational Complexity (ECCC) 2(56) (1995)
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)
Goldreich, O., Sudan, M.: Locally testable codes and PCPs of almost-linear length. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, Vancouver, Canada, November 16-19 (2002)
Halevy, S., Kushilevitz, E.: Distribution-free property-testing. SIAM J. Comput. 37(4), 1107–1138 (2007)
Jutla, C.S., Patthak, A.C., Rudra, A., Zuckerman, D.: Testing low-degree polynomials over prime fields. In: FOCS 2004: Proceedings of the Forty-Fifth Annual IEEE Symposium on Foundations of Computer Science, pp. 423–432. IEEE Computer Society Press, Los Alamitos (2004)
Kiwi, M.A.: Kiwi. Algebraic testing and weight distributions of codes. Theor. Comput. Sci. 3(299), 81–106 (2003)
Kaufman, T., Ron, D.: Testing polynomials over general fields. In: Proceedings of the Forty-fifthth Annual Symposium on Foundations of Computer Science, pp. 413–422 (2004)
Kindler, G., Safra, S.: Noise-resistant boolean-functions are juntas, April 17 (2003)
Kaufman, T., Sudan, M.: Sparse random linear codes are locally decodable and testable. In: FOCS, pp. 590–600. IEEE Computer Society Press, Los Alamitos (2007)
Kaufman, T., Sudan, M.: Algebraic property testing: the role of invariance. In: Ladner, R.E., Dwork, C. (eds.) STOC, pp. 403–412. ACM Press, New York (2008)
Meir, O.: Combinatorial construction of locally testable codes. In: Ladner, R.E., Dwork, C. (eds.) STOC, pp. 285–294. ACM Press, New York (2008)
Parnas, M., Ron, D., Rubinfeld, R.: Tolerant property testing and distance approximation. J. Comput. Syst. Sci. 72(6), 1012–1042 (2006)
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kopparty, S., Saraf, S. (2009). Tolerant Linearity Testing and Locally Testable Codes. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2009 2009. Lecture Notes in Computer Science, vol 5687. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-03685-9_45
Download citation
DOI: https://doi.org/10.1007/978-3-642-03685-9_45
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-03684-2
Online ISBN: 978-3-642-03685-9
eBook Packages: Computer ScienceComputer Science (R0)