Block-symmetric polynomials correlate with parity better than symmetric

We show that degree- d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree- d symmetric polynomials, if n ≥ cd^2 log d and d ∈ [0.995 · p^t - 1,p^t) for some t ≥ 1 and some c > 0 that depends only on p . For these infinitely many degrees, our result solves an open problem raised by a number of researchers including Alon Beigel (IEEE conference on computational complexity (CCC), pp 184–187, 2001 ). The only previous case for which this was known was d = 2 and p = 3 (Green in J Comput Syst Sci 69(1):28–44, 2004 ). The result is obtained through the development of a theory we call spectral analysis of symmetric correlation , which originated in works of Cai et al . (Math Syst Theory 29(3):245–258, 1996 ) and Green (Theory Comput Syst 32(4):453–466, 1999 ). In particular, our result follows from a detailed analysis of the correlation of symmetric polynomials, which is determined up to an exponentially small relative error when d = p^t-1 .