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We prove the "Most informative boolean function" conjecture of Courtade andKumar 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 byflipping 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 holdonly for balanced functions.

The "Most informative boolean function" conjecture holds for high noise