Block-symmetric polynomials correlate with parity better than symmetric

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 ≥ 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. The result is obtained through the development of a theory we call spectral analysis of symmetric correlation, which originated in works of Cai, Green, and Thierauf [CGT96,Gre99]. 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. *

Efficient resilient functions

“…The maximizing polynomials in [Gre04,GR10] are not symmetric. Moreover, the work [GKV17] has shown that for a large range of parameters, symmetric polynomials modulo 3 do not correlate best with parity (and are not even close). One of the families of polynomials that are shown to outperform symmetric in these works is that of block-symmetric polynomials, which are sums of symmetric polynomials on disjoint sets of variables.…”

“…However, naive conjectures regarding the optimality of block-symmetric or other families of polynomials fail, and we are not aware of any natural family of polynomials modulo 3 that is a candidate to maximizing correlation with parity. The only available evidence that symmetric polynomials correlate best with mod functions are computer experiments up to 10 variables reported in [GKV17].…”

Products of ideals and jet schemes

“…. , z Λi+1 for every 1 i n. The notion of λ-symmetric polynomials generalizes the notion of block-symmetric polynomials [2].…”