A short note on the joint entropy of n/2-wise independence

TL;DR

This paper establishes tight lower bounds on the joint entropy of n/2-wise independent Bernoulli variables and extends bounds for general k-wise independence, advancing understanding in information theory and combinatorics.

Contribution

It provides the first tight lower bounds for joint entropy under n/2-wise independence and introduces new bounds for general k-wise independence using Fourier analysis methods.

Findings

Proves tight lower bounds for joint entropy of n/2-wise independent Bernoulli variables.

Extends lower bounds to general k-wise independence using Fourier techniques.

Makes partial progress on a problem posed by Gavinsky and Pudlák.

Abstract

In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitsky's Fourier proof of the `LP bound' on error correcting codes. This counts as partial progress on a problem asked by Gavinsky and Pudl\'ak.