On XOR lemmas for the weight of polynomial threshold functions

Abstract

A multilinear polynomial p is said to sign-represent a Boolean function f:{−1,1}n→{−1,1} if f(x)=sgn(p(x)) for all x∈{−1,1}n. In this paper, we consider the length and weight of polynomials sign-representing Boolean functions of the form ⊕kf, the XOR of k copies of f on disjoint sets of variables. Firstly, we show that for an infinite family of functions f, a naive construction does not yield a shortest polynomial sign-representing ⊕kf. More precisely, we give a construction of polynomials sign-representing ⊕kANDn whose length is strictly smaller than the k-th power of the minimum length of a polynomial sign-representing ANDn, for every k≥2 and n≥2 except for k=n=2. Previously, such polynomials were known only for n=2 (Sezener and Oztop, 2015). A similar result for the weight is also provided. Secondly, we introduce a parameter vf⁎ of a Boolean function f and show that the k-th root of the minimum weight of a polynomial sign-representing ⊕kf converges between vf⁎ and (vf⁎)2 as k goes to infinity.