## Abstract

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 \({n \ge cd^2 {\rm log} d}\) and \({d \in [0.995 \cdot p^t - 1,p^t)}\) for some \({t \ge 1}\) 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 \({d = p^t-1}\).

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Green, F., Kreymer, D. & Viola, E. Block-symmetric polynomials correlate with parity better than symmetric.
*comput. complex.* **26, **323–364 (2017). https://doi.org/10.1007/s00037-017-0153-3

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### Keywords

- polynomial
- correlation
- parity
- mod
*m* - symmetric
- degree

### Subject classification

- 68Q99