On the Most Informative Boolean Functions of the Very Noisy Channel

Let $X^n$ be a uniformly distributed $n$-dimensional binary vector, and $Y^n$ be the result of passing $X^n$ through a binary symmetric channel (BSC) with crossover probability $\alpha$. A recent conjecture postulated by Courtade and Kumar states that for any Boolean function $f:\{0,1{\}}^n\to \{0,1\}$, $I(f(X^n);Y^n)\le 1-H(\alpha )$. Although the conjecture has been proved to be true in the dimension-free high noise regime by Samorodnitsky, here we present a calculus-based approach to show a dimension-dependent result by examining the second derivative of $H(\alpha )-H(f(X^n)\mathrm{\mid }{Y}^n)$ at $\alpha =1\mathrm{/}2$. Along the way, we show that the dictator function is the most informative function in the high noise regime.