Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates

Block-symmetric polynomials correlate with parity better than symmetric

“…where r(k) is the value of the polynomial on inputs of Hamming weight k. The advantage of our above Expression (5) for β over the previous expressions in [ CGT96,Gre99] is that ours depends only on the values of the symmetric polynomial at the first D Hamming weights. As we show in Lemma 3.4, this grants us complete freedom in the t − for > 1 is reduced to the case = 1.)…”

“…Our main results are obtained by developing a theory we call spectral analysis of symmetric correlation, which originated in works of Cai, Green, and Thierauf [CGT96,Gre99]. Before describing it, we mention a useful way to name symmetric polynomials independently of the number n of variables.…”

“…In §2 we obtain our spectral formulation which refines [CGT96,Gre99], providing in particular an expression for β. Various bounds on β are obtained in §3, establishing our main Theorem 1.2.…”

“…The proof of Lemma 2.1 uses the following lemma from [Gre99]. For completeness, we remark that the latter is proved by first rewriting the correlation (cf.…”

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AbstractWe 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.

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