New correlation bounds for GF(2) polynomials using Gowers uniformity

We study the correlation between low-degree GF(2) polynomials p and explicit functions. Our main results are the following: I We prove that the Modm function on n bits has correlation at most exp i i­ i n/4 d ¢¢ with any GF(2) polynomial of degree d, for any fixed odd integer m. This improves on the previous exp i i­ i n/8 d ¢¢ bound by Bourgain (C. R. Acad. Sci. Paris,

[1]  Alexander Healy,et al.  Randomness-E-cient Sampling within NC 1 , 2007 .

[2]  Alexander Healy,et al.  Randomness-Efficient Sampling within NC1 , 2006, computational complexity.

[3]  Emanuele Viola,et al.  Fooling Parity Tests with Parity Gates , 2004, APPROX-RANDOM.

[4]  W. T. Gowers,et al.  A new proof of Szemerédi's theorem , 2001 .

[5]  Fan Chung Graham,et al.  Communication Complexity and Quasi Randomness , 1993, SIAM J. Discret. Math..

[6]  Ben Green,et al.  AN INVERSE THEOREM FOR THE GOWERS $U^3(G)$ NORM , 2008, Proceedings of the Edinburgh Mathematical Society.

[7]  J. Bourgain.  Estimation of certain exponential sums arising in complexity theory , 2005 .

[8]  Noga Alon,et al.  Testing Low-Degree Polynomials over GF(2( , 2003, RANDOM-APPROX.

[9]  Alex Samorodnitsky,et al.  Low-degree tests at large distances , 2006, STOC '07.

[10]  Noga Alon,et al.  Lower bounds for approximations by low degree polynomials over Z/sub m/ , 2001, Proceedings 16th Annual IEEE Conference on Computational Complexity.

[11]  Pavel Pudlák,et al.  Threshold circuits of bounded depth , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[12]  Howard Straubing,et al.  Bounds on an exponential sum arising in Boolean circuit complexity , 2005 .

[13]  Frederic Green,et al.  The correlation between parity and quadratic polynomials mod 3 , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[14]  Kristoffer Arnsfelt Hansen.  Lower Bounds for Circuits with Few Modular Gates using Exponential Sums , 2006, Electron. Colloquium Comput. Complex..

[15]  Moni Naor,et al.  Small-Bias Probability Spaces: Efficient Constructions and Applications , 1993, SIAM J. Comput..

[16]  Luca Trevisan,et al.  Gowers uniformity, influence of variables, and PCPs , 2005, STOC '06.

[17]  Noga Alon,et al.  Simple Construction of Almost k-wise Independent Random Variables , 1992, Random Struct. Algorithms.

[18]  Noam Nisan,et al.  On Yao's XOR-Lemma , 1995, Electron. Colloquium Comput. Complex..

[19]  W. T. Gowers,et al.  A NEW PROOF OF SZEMER ´ EDI'S THEOREM , 2001 .

[20]  S. Vadhan,et al.  The complexity of hardness amplification and derandomization , 2006 .

[21]  Johan Håstad,et al.  On the power of small-depth threshold circuits , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[22]  Emanuele Viola,et al.  Pseudorandom bits for constant depth circuits with few arbitrary symmetric gates , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[23]  W. T. Gowers,et al.  A New Proof of Szemerédi's Theorem for Arithmetic Progressions of Length Four , 1998 .

[24]  Roman Smolensky,et al.  Algebraic methods in the theory of lower bounds for Boolean circuit complexity , 1987, STOC.

[25]  Atri Rudra,et al.  Testing low-degree polynomials over prime fields , 2004, 45th Annual IEEE Symposium on Foundations of Computer Science.