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Robust Multiplication-Based Tests for Reed-Muller Codes

Authors Prahladh Harsha, Srikanth Srinivasan

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  • Polynomials over finite fields
  • Schwartz-Zippel lemma
  • Low degree testing
  • Low degree long code

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Abstract

We consider the following multiplication-based tests to check if a given function f: F^n_q -> F_q is the evaluation of a degree-d polynomial over F_q for q prime.

Test_{e,k}: Pick P_1,...,P_k independent random degree-e polynomials and accept iff the function f P_1 ... P_k is the evaluation of a degree-(d + ek) polynomial.

We prove the robust soundness of the above tests for large values of e, answering a question of Dinur and Guruswami (FOCS 2013). Previous soundness analyses of these tests were known only for the case when either e = 1 or k = 1. Even for the case k = 1 and e > 1, earlier soundness analyses were not robust. 

We also analyze a derandomized version of this test, where (for example) the polynomials P_1 ,... , P_k can be the same random polynomial P. This generalizes a result of Guruswami et al. (STOC 2014).

One of the key ingredients that go into the proof of this robust soundness is an extension of the standard Schwartz-Zippel lemma over general finite fields F_q, which may be of independent interest.

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Prahladh Harsha and Srikanth Srinivasan. Robust Multiplication-Based Tests for Reed-Muller Codes. In 36th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 65, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016) https://doi.org/10.4230/LIPIcs.FSTTCS.2016.17

Author Details

Prahladh Harsha
Srikanth Srinivasan

References

  1. Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing Reed-Muller codes. IEEE Trans. Inform. Theory, 51(11):4032-4039, 2005. (Preliminary version in 7th RANDOM, 2003). URL: http://dx.doi.org/10.1109/TIT.2005.856958.
  2. Boaz Barak, Parikshit Gopalan, Johan Håstad, Raghu Meka, Prasad Raghavendra, and David Steurer. Making the long code shorter. SIAM J. Comput., 44(5):1287-1324, 2015. (Preliminary version in 53rd FOCS, 2012). http://arxiv.org/abs/1111.0405, URL: http://dx.doi.org/10.1137/130929394.
  3. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of Reed-Muller codes. In Proc. 51st IEEE Symp. on Foundations of Comp. Science (FOCS), pages 488-497, 2010. http://arxiv.org/abs/0910.0641, URL: http://dx.doi.org/10.1109/FOCS.2010.54.
  4. Irit Dinur and Venkatesan Guruswami. PCPs via the low-degree long code and hardness for constrained hypergraph coloring. Israel Journal of Mathematics, 209:611-649, 2015. (Preliminary version in 54th FOCS, 2013). URL: http://dx.doi.org/10.1007/s11856-015-1231-3.
  5. Venkat Guruswami, Prahladh Harsha, Johan Håstad, Srikanth Srinivasan, and Girish Varma. Super-polylogarithmic hypergraph coloring hardness via low-degree long codes. In Proc. 46th ACM Symp. on Theory of Computing (STOC), pages 614-623, 2014. http://arxiv.org/abs/1311.7407, URL: http://dx.doi.org/10.1145/2591796.2591882.
  6. Elad Haramaty, Amir Shpilka, and Madhu Sudan. Optimal testing of multivariate polynomials over small prime fields. SIAM J. Comput., 42(2):536-562, 2013. (Preliminary version in 52nd FOCS, 2011). URL: http://dx.doi.org/10.1137/120879257.
  7. Sangxia Huang. 2^(log N)^1/10-o(1) hardness for hypergraph coloring. (manuscript), 2015. URL: http://arxiv.org/abs/1504.03923.
  8. Tadao Kasami, Shu Lin, and W. Wesley Peterson. Polynomial codes. IEEE Trans. Inform. Theory, 14(6):807-814, 1968. URL: http://dx.doi.org/10.1109/TIT.1968.1054226.
  9. Subhash Khot and Rishi Saket. Hardness of coloring 2-colorable 12-uniform hypergraphs with 2^(log n)^Ω(1) colors. In Proc. 55th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 206-215, 2014. URL: http://dx.doi.org/10.1109/FOCS.2014.30.
  10. Girish Varma. Reducing uniformity in Khot-Saket hypergraph coloring hardness reductions. Chicago J. Theor. Comput. Sci., 2015(3):1-8, 2015. http://arxiv.org/abs/1408.0262, URL: http://dx.doi.org/10.4086/cjtcs.2015.003.
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