Local Decoding and Testing of Polynomials over Grids

Authors Srikanth Srinivasan, Madhu Sudan

Thumbnail PDF


  • Filesize: 0.55 MB
  • 14 pages

Document Identifiers

Author Details

Srikanth Srinivasan
Madhu Sudan

Cite AsGet BibTex

Srikanth Srinivasan and Madhu Sudan. Local Decoding and Testing of Polynomials over Grids. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 26:1-26:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


The well-known DeMillo-Lipton-Schwartz-Zippel lemma says that n-variate polynomials of total degree at most d over grids, i.e. sets of the form A_1 \times A_2 \times \cdots \times A_n, form error-correcting codes (of distance at least 2^{-d} provided \min_i\{|A_i|\}\geq 2). In this work we explore their local decodability and local testability. While these aspects have been studied extensively when A_1 = \cdots = A_n = \F_q are the same finite field, the setting when A_i's are not the full field does not seem to have been explored before. In this work we focus on the case A_i = {0,1} for every i. We show that for every field (finite or otherwise) there is a test whose query complexity depends only on the degree (and not on the number of variables). In contrast we show that decodability is possible over fields of positive characteristic (with query complexity growing with the degree of the polynomial and the characteristic), but not over the reals, where the query complexity must grow with $n$. As a consequence we get a natural example of a code (one with a transitive group of symmetries) that is locally testable but not locally decodable. Classical results on local decoding and testing of polynomials have relied on the 2-transitive symmetries of the space of low-degree polynomials (under affine transformations). Grids do not possess this symmetry: So we introduce some new techniques to overcome this handicap and in particular use the hypercontractivity of the (constant weight) noise operator on the Hamming cube.
  • Property testing
  • Coding theory
  • Low-degree testing
  • Local decoding
  • Local testing


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Noga Alon, Tali Kaufman, Michael Krivelevich, Simon Litsyn, and Dana Ron. Testing reed-muller codes. IEEE Trans. Information Theory, 51(11):4032-4039, 2005. URL: http://dx.doi.org/10.1109/TIT.2005.856958.
  2. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: http://dx.doi.org/10.1145/278298.278306.
  3. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: A new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: http://dx.doi.org/10.1145/273865.273901.
  4. Donald Beaver and Joan Feigenbaum. Hiding instances in multioracle queries. In C. Choffrut and T. Lengauer, editors, Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, pages 37-48, Rouen, France, 22-24 February 1990. Springer. Lecture Notes in Computer Science, Volume 415. Google Scholar
  5. Michael Ben-Or, Shafi Goldwasser, and Avi Wigderson. Completeness theorems for non-cryptographic fault-tolerant distributed computation (extended abstract). In Janos Simon, editor, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, May 2-4, 1988, Chicago, Illinois, USA, pages 1-10. ACM, 1988. URL: http://dx.doi.org/10.1145/62212.62213.
  6. Eli Ben-Sasson, Prahladh Harsha, and Sofya Raskhodnikova. Some 3cnf properties are hard to test. SIAM J. Comput., 35(1):1-21, 2005. URL: http://dx.doi.org/10.1137/S0097539704445445.
  7. Arnab Bhattacharyya, Swastik Kopparty, Grant Schoenebeck, Madhu Sudan, and David Zuckerman. Optimal testing of reed-muller codes. In 51th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2010, October 23-26, 2010, Las Vegas, Nevada, USA, pages 488-497. IEEE Computer Society, 2010. URL: http://dx.doi.org/10.1109/FOCS.2010.54.
  8. Manuel Blum, Ashok K. Chandra, and Mark N. Wegman. Equivalence of free boolean graphs can be decided probabilistically in polynomial time. Inf. Process. Lett., 10(2):80-82, 1980. URL: http://dx.doi.org/10.1016/S0020-0190(80)90078-2.
  9. Manuel Blum, Michael Luby, and Ronitt Rubinfeld. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993. Google Scholar
  10. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL: http://dx.doi.org/10.1016/0020-0190(78)90067-4.
  11. Irit Dinur. The PCP theorem by gap amplification. J. ACM, 54(3):12, 2007. URL: http://dx.doi.org/10.1145/1236457.1236459.
  12. Oded Goldreich and Madhu Sudan. Locally testable codes and pcps of almost-linear length. J. ACM, 53(4):558-655, 2006. URL: http://dx.doi.org/10.1145/1162349.1162351.
  13. Charanjit S. Jutla, Anindya C. Patthak, Atri Rudra, and David Zuckerman. Testing low-degree polynomials over prime fields. Random Struct. Algorithms, 35(2):163-193, 2009. URL: http://dx.doi.org/10.1002/rsa.20262.
  14. Jonathan Katz and Luca Trevisan. On the efficiency of local decoding procedures for error-correcting codes. In F. Frances Yao and Eugene M. Luks, editors, Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, May 21-23, 2000, Portland, OR, USA, pages 80-86. ACM, 2000. URL: http://dx.doi.org/10.1145/335305.335315.
  15. Tali Kaufman and Dana Ron. Testing polynomials over general fields. SIAM J. Comput., 36(3):779-802, 2006. URL: http://dx.doi.org/10.1137/S0097539704445615.
  16. Tali Kaufman and Madhu Sudan. Algebraic property testing: the role of invariance. In Cynthia Dwork, editor, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, Victoria, British Columbia, Canada, May 17-20, 2008, pages 403-412. ACM, 2008. URL: http://dx.doi.org/10.1145/1374376.1374434.
  17. John Y. Kim and Swastik Kopparty. Decoding reed-muller codes over product sets. In Ran Raz, editor, 31st Conference on Computational Complexity, CCC 2016, May 29 to June 1, 2016, Tokyo, Japan, volume 50 of LIPIcs, pages 11:1-11:28. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. URL: http://dx.doi.org/10.4230/LIPIcs.CCC.2016.11.
  18. Richard Lipton. New directions in testing. In Distributed Computing and Cryptography, volume 2 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science, pages 191-202. AMS, 1991. Google Scholar
  19. Carsten Lund, Lance Fortnow, Howard J. Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. J. ACM, 39(4):859-868, 1992. URL: http://dx.doi.org/10.1145/146585.146605.
  20. D. E. Muller. Application of Boolean algebra to switching circuit design and to error detection. IEEE Transactions on Computers, 3:6-12, 1954. Google Scholar
  21. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL: http://dx.doi.org/10.1007/BF02579206.
  22. Yury Polyanskiy. Hypercontractivity of spherical averages in Hamming space. CoRR, abs/1309.3014, 2013. Google Scholar
  23. Alexander A. Razborov. On the method of approximations. In David S. Johnson, editor, Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, Seattle, Washigton, USA, pages 167-176. ACM, 1989. URL: http://dx.doi.org/10.1145/73007.73023.
  24. Irving S. Reed. A class of multiple-error-correcting codes and the decoding scheme. IEEE Transactions on Information Theory, 4:38-49, 1954. Google Scholar
  25. Ronitt Rubinfeld and Madhu Sudan. Robust characterizations of polynomials with applications to program testing. SIAM Journal on Computing, 25(2):252-271, April 1996. Google Scholar
  26. Jacob T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4):701-717, 1980. URL: http://dx.doi.org/10.1145/322217.322225.
  27. Adi Shamir. How to share a secret. Commun. ACM, 22(11):612-613, 1979. URL: http://dx.doi.org/10.1145/359168.359176.
  28. Adi Shamir. Ip=pspace. In 31st Annual Symposium on Foundations of Computer Science, St. Louis, Missouri, USA, October 22-24, 1990, Volume I, pages 11-15. IEEE Computer Society, 1990. URL: http://dx.doi.org/10.1109/FSCS.1990.89519.
  29. Srikanth Srinivasan and Madhu Sudan. Local decoding and testing of polynomials over grids. CoRR, abs/1709.06036, 2017. URL: http://arxiv.org/abs/1709.06036.
  30. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Edward W. Ng, editor, Symbolic and Algebraic Computation, EUROSAM '79, An International Symposiumon Symbolic and Algebraic Computation, Marseille, France, June 1979, Proceedings, volume 72 of Lecture Notes in Computer Science, pages 216-226. Springer, 1979. URL: http://dx.doi.org/10.1007/3-540-09519-5_73.
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail