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 theModm function on n bits has correlation at most exp ( −Ω ( n/4d )) with any GF (2) polynomial of degree d, for any fixed odd integer m. This improves on the previous exp ( −Ω ( n/8d )) bound by Bourgain (C. R. Acad. Sci. Paris, 2005) and Green et al. (C. R. Acad. Sci. Paris, 2005). II We exhibit a polynomial-time computable function on n bits that has correlation at most exp ( −Ω ( n/2d )) with any GF (2) polynomial of degree d. Previous to our work the best correlation bound for an explicit function was exp ( −Ω (