  We prove the "Most informative boolean function" conjecture of Courtade and
Kumar for high noise $\epsilon \ge 1/2 - \delta$, for some absolute constant
$\delta > 0$.
  Namely, if $X$ is uniformly distributed in $\{0,1\}^n$ and $Y$ is obtained by
flipping each coordinate of $X$ independently with probability $\epsilon$,
then, provided $\epsilon \ge 1/2 - \delta$, for any boolean function $f$ holds
$I(f(X);Y) \le 1 - H(\epsilon)$. This conjecture was previously known to hold
only for balanced functions.
