Majority optimality in NICD with erasures for p<1/2

Determine whether, in the binary erasure Non-Interactive Correlation Distillation (NICD) model with erasure rate p<1/2, the quantity Φ_p(f) = E_z[|f(z)|], where z ∈ {−1,0,1}^n has independent coordinates with P(z_i=±1)=p/2 and P(z_i=0)=1−p and where f: {−1,1}^n → {−1,1} is evaluated at z via its unique multilinear extension, is maximized over all unbiased Boolean functions by the majority function Maj_n.

Background

The paper studies the Non-Interactive Correlation Distillation (NICD) problem under a binary erasure model: each coordinate is independently erased with probability 1−p, and otherwise revealed as ±1 with probability p/2 each. For an unbiased Boolean function f, the objective is Φ_p(f) = E[|f(z)|], where f is evaluated at the erasure sample z via its multilinear extension.

The Simons Institute open problems list recorded the question of whether, for p<1/2, the majority function maximizes Φ_p among unbiased Boolean functions. The present note provides a concrete counterexample at n=5 and p=0.40, showing that the general statement is false. The authors also prove that for each fixed odd n, majority is optimal for sufficiently small p>0, partially clarifying the regime near p=0.