Testing polynomials over general fields

SIAM Journal on Computing

Volumn 36, Issue 3, 2006, Pages 779-802

  Kaufman, Tali ^a   Ron, Dana ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b TEL AVIV UNIVERSITY   (Israel)

Author keywords

Polynomials; Reed Muller codes; Testing

Indexed keywords

KNOWLEDGE GAP; REED-MULLER CODES; SUBSPACES;

BOUNDARY CONDITIONS; ENCODING (SYMBOLS); INTEGER PROGRAMMING;

POLYNOMIALS;

EID: 34250862184     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539704445615     Document Type: Article

References (32)

1. N. ALON, M. KRIVELEVICH, T. KAUFMAN, S. LITSYN, AND D. RON, Testing Reed-Muller codes, IEEE Trans. Inform. Theory, 51 (2005), pp. 4032-4038.
2. N. ALON, M. KRIVELEVICH, T. KAUFMAN, AND D. RON, Testing triangle-freeness in general graphs, in Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, SIAM, Philadelphia, 2006.
3. S. ARORA, Probabilistic Checking of Proofs and the Hardness of Approximation Problems, Ph.D. thesis, UC Berkeley, Berkeley, CA, 1994.
4. S. ARORA AND M. SUDAN, Improved low-degree testing and its applications, in Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, ACM, New York, 1997, pp. 485-495.
5. E. F. ASSMUS, JR., AND J. D. KEY, Designs and Their Codes, Cambridge Tracts in Math. 103, Cambridge University Press, Cambridge, UK, 1992.
6. L. BABAI, L. FORTNOW, L. LEVIN, AND M. SZEGEDY, Checking computations in polylogarithmic time, in Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 21-31.
7. L. BABAI, L. FORTNOW, AND C. LUND, Non-deterministic exponential time has two-prover interactive protocols, Comput. Complexity, 1 (1991), pp. 3-40.
8. L. BABAI, A. SHPILKA, AND D. STEFANKOVIC, Locally testable cyclic codes, IEEE Trans. Inform. Theory, 51 (2005), pp. 2849-2858.
9. M. BELLARE, D. COPPERSMITH, J. HÅSTAD, M. KIWI, AND M. SUDAN, Linearity testing over characteristic two, IEEE Trans. Inform. Theory, 42 (1996), pp. 1781-1795.
10. M. BELLARE, S. GOLDWASSER, C. LUND, AND A. RUSSELL, Efficient probabilistically checkable proofs and applications to approximation, in Proceedings of the Twenty-Fifth Annual ACM Symposium on the Theory of Computing, ACM, New York, 1993, pp. 294-304.
11. M. BELLARE AND M. SUDAN, Improved non-approximability results, in Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, ACM, New York, 1994, pp. 184-193.
12. E. BEN-SASSON, O. GOLDREICH, P. HARSHA, M. SUDAN, AND S. VADHAN, Robust PCPs of proximity, shorter PCPs and applications to coding, in Proceedings of the Thirty-Sixth Annual ACM Symposium on the Theory of Computing, ACM, New York, 2004, pp. 1-10.
13. E. BEN-SASSON, P. HARSHA, AND S. RASKHODNIKOVA, Some 3CNF properties are hard to test, SIAM J. Comput., 35 (2005), pp. 1-21.
14. E. BEN-SASSON, M. SUDAN, S. VADHAN, AND A. WIGDERSON, Derandomizing low degree tests via epsilon-biased spaces, in Proceedings of the Thirty-Fifth Annual ACM Symposium on the Theory of Computing, ACM, New York, 2003, pp. 612-621.
15. M. BLUM, M. LUBY, AND R. RUBINFELD, Self-testing/correcting with applications to numerical problems, J. Comput. System Sci., 47 (1993), pp. 549-595.
16. P. DING AND J. D. KEY, Minimum-weight codewords as generators of generalized Reed-Muller codes, IEEE Trans. Inform. Theory, 46 (2000), pp. 2152-2157.
17. I. DINUR, The PCP theorem via gap amplification, in Proceedings of the Thirty-Eighth Annual ACM Symposium on the Theory of Computing, ACM, New York, 2006, pp. 241-250.
18. U. FEIGE, S. GOLDWASSER, L. LOVÁSZ, S. SAFRA, AND M. SZEGEDY, Approximating clique is almost NP-complete, JACM, (1996), pp. 268-292.
19. K. FRIEDL AND M. SUDAN, Some improvements to total degree tests, in Proceedings of the 3rd Annual Israel Symposium on Theory of Computing and Systems, Tel Aviv, Israel, 1995, pp. 190-198; corrected version available online at http://theory.lcs.mit.edu/~madhu/papers/friedl.ps.
20. P. GEMMELL, R. LIPTON, R. RUBINFELD, M. SUDAN, AND A. WIGDERSON, Self-testing/correcting for polynomials and for approximate functions, in Proceedings of the Twenty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 1991, pp. 32-42.
21. O. GOLDREICH, S. GOLDWASSBR, AND D. RON, Property testing and its connection to learning and approximation, JACM, 45 (1998), pp. 653-750.
22. O. GOLDREICH AND D. RON, A sublinear bipartite tester for bounded degree graphs, Combinatorics, 19 (1999), pp. 335-373.
23. O. GOLDREICH AND D. RON, Property testing in bounded degree graphs, Algorithmica, 32 (2002), pp. 302-343.
24. O. GOLDREICH AND M. SUDAN, Locally testable codes and PCPs of almost-linear length, in Proceedings of the Forty-Third Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2002, pp. 13-22.
25. C. S. JUTLA, A. C. PATTHAK, A. RUDRA, AND D. ZUCKERMAN, Testing low-degree polynomials over prime fields, in Proceedings of the Forty-Fifth Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2004, pp. 423-432.
26. T. KAUFMAN, M. KRIVELEVICH, AND D. RON, Tight bounds for testing bipartiteness in general graphs, SIAM J. Comput., 33 (2004), pp. 1441-1483.
27. T. KAUFMAN AND S. LITSYN, Almost orthogonal linear codes are locally testable, in Proceedings of the Forty-Sixth Annual IEEE Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2005, pp. 317-326.
28. F. J. MACWILLIAMS AND N. J. A. SLOANE, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977.
29. A. POLISHCHUK AND D. SPIELMAN, Nearly-linear size holographic proofs, in Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, ACM, New York, 1994, pp. 194-203.
30. R. RUBINFELD AND M. SUDAN, Robust characterization of polynomials with applications to program testing, SIAM J. Comput., 25 (1996), pp. 252-271.
31. A. SHPILKA AND A. WIGDERSON, Derandomizing homomorphism testing in general groups, in Proceedings of the Thirty-Sixth Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 427-435.
32. M. SUDAN, Efficient Checking of Polynomials and Proofs and the Hardness of Approximation Problems, Lecture Notes in Comput. Sci. 1001, Springer-Verlag, Berlin, 1996.