  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.
