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 $f:{0,1}^nto{0,1}$, $I(f(X^n);Y^n)le 1-H(alpha)$. 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 $H(alpha)-H(f(X^n)|Y^n)$ at $alpha=1/2$. Along the way, we show that the dictator function is the most informative function in the high noise regime.