# On the Courtade-Kumar conjecture for certain classes of Boolean functions

## TL;DR

This paper proves the Courtade-Kumar conjecture for specific classes of Boolean functions, showing that mutual information between the function output and channel output is bounded by the binary entropy, for all input sizes and error probabilities.

## Contribution

The authors establish the conjecture for certain classes of Boolean functions, extending the understanding of mutual information bounds in binary symmetric channels.

## Key findings

- Proved the conjecture for specific Boolean function classes.
- Mutual information is bounded by 1 - H(p) for all n ≥ 2 and 0 ≤ p ≤ 0.5.
- Results hold universally across different input sizes and error probabilities.

## Abstract

We prove the Courtade-Kumar conjecture, for certain classes of $n$-dimensional Boolean functions, $\forall n\geq 2$ and for all values of the error probability of the binary symmetric channel, $\forall 0 \leq p \leq \frac{1}{2}$. Let $\mathbf{X}=[X_1...X_n]$ be a vector of independent and identically distributed Bernoulli$(\frac{1}{2})$ random variables, which are the input to a memoryless binary symmetric channel, with the error probability in the interval $0 \leq p \leq \frac{1}{2}$, and $\mathbf{Y}=[Y_1...Y_n]$ the corresponding output. Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be an $n$-dimensional Boolean function. Then, the Courtade-Kumar conjecture states that the mutual information $\operatorname{MI}(f(\mathbf{X}),\mathbf{Y}) \leq 1-\operatorname{H}(p)$, where $\operatorname{H}(p)$ is the binary entropy function.