A Lower Bound on the Constant in the Fourier Min-Entropy/Influence Conjecture

TL;DR

This paper introduces a new Boolean function construction that achieves the highest known min-entropy/influence ratio, providing the best lower bound to date for the Fourier min-entropy/influence conjecture.

Contribution

It presents a novel Boolean function construction that improves the lower bound on the Fourier min-entropy/influence conjecture.

Findings

Achieved a min-entropy/influence ratio of approximately 2.8444.

Constructed a 30-variable Boolean function with the highest known ratio.

Established the best known lower bound on the conjecture's universal constant.

Abstract

We describe a new construction of Boolean functions. A specific instance of our construction provides a 30-variable Boolean function having min-entropy/influence ratio to be $128/45 \approx 2.8444$ which is presently the highest known value of this ratio that is achieved by any Boolean function. Correspondingly, $128/45$ is also presently the best known lower bound on the universal constant of the Fourier min-entropy/influence conjecture.