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.
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References
Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., Ron, D.: Testing Reed-Muller codes. IEEE Transactions on Information Theory 51(11), 4032–4039 (2005)
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)
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)
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)
Ben-Sasson, E., Harsha, P., Raskhodnikova, S.: Some 3CNF properties are hard to test. SICOMP: SIAM Journal on Computing 35 (2005)
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)
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)
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)
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
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)
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)
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)
Kaufman, T., Ron, D.: Testing polynomials over general fields. SIAM Journal of Computing 36(3), 779–802 (2006)
Kaufman, T., Sudan, M.: Algebraic property testing: The role of invariance. Electronic Colloquium on Computational Complexity (ECCC) 14(111) (2007)
Rubinfeld, R., Sudan, M.: Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing 25(2), 252–271 (1996)
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)
Viola, E., Wigderson, A.: Norms, xor lemmas, and lower bounds for polynomials and protocols. Theory of Computing 4(1), 137–168 (2008)
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Ron-Zewi, N., Sudan, M. (2012). A New Upper Bound on the Query Complexity for Testing Generalized Reed-Muller codes. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_54
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DOI: https://doi.org/10.1007/978-3-642-32512-0_54
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