Advertisement

Abstract

Over a finite field \({\mathbb{F}}_q\) the (n,d,q)-Reed-Muller code is the code given by evaluations of n-variate polynomials of total degree at most d on all points (of \({\mathbb{F}}_q^n\)). The task of testing if a function \(f:{\mathbb{F}}_q^n \to {\mathbb{F}}_q\) is close to a codeword of an (n,d,q)-Reed-Muller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all Reed-Muller codewords and rejecting words that are δ-far from the code with probability Ω(δ). (In this work we allow the constant in the Ω to depend on d.)

For codes over a prime field \({\mathbb F}_q\) the optimal query complexity is well-known and known to be Θ(q ⌈(d + 1)/(q − 1)⌉), and the test consists of testing if f is a degree d polynomial on a randomly chosen (⌈(d + 1)/(q − 1) ⌉)-dimensional affine subspace of \({\mathbb{F}}_q^n\). If q is not a prime, then the above quantity remains a lower bound, whereas the previously known upper bound grows to O(q ⌈(d + 1)/(q − q/p) ⌉) where p is the characteristic of the field \({\mathbb{F}}_q\). In this work we give a new upper bound of (c q)(d + 1)/q on the query complexity, where c is a universal constant. Thus for every p and sufficiently large q this bound improves over the previously known bound by a polynomial factor.

In the process we also give new upper bounds on the “spanning weight” of the dual of the Reed-Muller code (which is also a Reed-Muller code). The spanning weight of a code is the smallest integer w such that codewords of Hamming weight at most w span the code. The main technical contribution of this work is the design of tests that test a function by not querying its value on an entire subspace of the space, but rather on a carefully chosen (algebraically nice) subset of the points from low-dimensional subspaces.

Keywords

Linear Property Total Degree Query Complexity Dual Code Muller 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [AKK+05]
    Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing Reed-Muller codes. IEEE Transactions on Information Theory 51(11), 4032–4039 (2005)MathSciNetCrossRefGoogle Scholar
  2. [ALM+98]
    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)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [BGH+11]
    Barak, B., Gopalan, P., Håstad, J., Meka, R., Raghavendra, P., Steurer, D.: Making the long code shorter, with applications to the unique games conjecture. CoRR, abs/1111.0405 (2011)Google Scholar
  4. [BGM+11]
    Ben-Sasson, E., Grigorescu, E., Maatouk, G., Shpilka, A., Sudan, M.: On Sums of Locally Testable Affine Invariant Properties. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 400–411. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  5. [BHR05]
    Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SICOMP: SIAM Journal on Computing  35 (2005)Google Scholar
  6. [BKS+10]
    Bhattacharyya, A., Kopparty, S., Schoenebeck, G., Sudan, M., Zuckerman, D.: Optimal testing of Reed-Muller codes. In: FOCS, pp. 488–497. IEEE Computer Society (2010)Google Scholar
  7. [BS11]
    Ben-Sasson, E., Sudan, M.: Limits on the Rate of Locally Testable Affine-Invariant Codes. In: Goldberg, L.A., Jansen, K., Ravi, R., Rolim, J.D.P. (eds.) APPROX/RANDOM 2011. LNCS, vol. 6845, pp. 412–423. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  8. [DK00]
    Ding, P., Key, J.D.: Minimum-weight codewords as generators of generalized Reed-Muller codes. IEEE Transactions on Information Theory 46(6), 2152–2158 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [FS95]
    Friedl, K., Sudan, M.: Some improvements to total degree tests. In: Proceedings of the 3rd Annual Israel Symposium on Theory of Computing and Systems, January 4-6, pp. 190–198. IEEE Computer Society, Washington, DC (1995) Corrected version available online at, http://people.csail.mit.edu/madhu/papers/friedl.ps CrossRefGoogle Scholar
  10. [GKS09]
    Grigorescu, E., Kaufman, T., Sudan, M.: Succinct Representation of Codes with Applications to Testing. In: Dinur, I., Jansen, K., Naor, J., Rolim, J.D.P. (eds.) APPROX/RANDOM 2009. LNCS, vol. 5687, pp. 534–547. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. [HSS11]
    Haramaty, E., Shpilka, A., Sudan, M.: Optimal testing of multivariate polynomials over small prime fields. In: Ostrovsky, R. (ed.) FOCS, pp. 629–637. IEEE (2011)Google Scholar
  12. [JPRZ09]
    Jutla, C.S., Patthak, A.C., Rudra, A., Zuckerman, D.: Testing low-degree polynomials over prime fields. Random Struct. Algorithms 35(2), 163–193 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. [KR06]
    Kaufman, T., Ron, D.: Testing polynomials over general fields. SIAM Journal of Computing 36(3), 779–802 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  14. [KS07]
    Kaufman, T., Sudan, M.: Algebraic property testing: The role of invariance. Electronic Colloquium on Computational Complexity (ECCC) 14(111) (2007)Google Scholar
  15. [RS96]
    Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  16. [RS12]
    Ron-Zewi, N., Sudan, M.: A new upper bound on the query complexity for testing generalized Reed-Muller codes. Electronic Colloquium on Computational Complexity (ECCC) 19, 46 (2012)Google Scholar
  17. [VW08]
    Viola, E., Wigderson, A.: Norms, xor lemmas, and lower bounds for polynomials and protocols. Theory of Computing 4(1), 137–168 (2008)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Noga Ron-Zewi
    • 1
  • Madhu Sudan
    • 2
  1. 1.Department of Computer ScienceTechnionHaifaIsrael
  2. 2.Microsoft Research New EnglandCambridgeUSA

Personalised recommendations