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

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.