Which Boolean Functions are Most Informative?

Abstract: We introduce a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let $X^n$ be i.i.d. Bernoulli(1/2), and let $Y^n$ be the result of passing $X^n$ through a memoryless binary symmetric channel with crossover probability $\alpha$. For any Boolean function $b:{0,1}^n\rightarrow {0,1}$, we conjecture that $I(b(X^n);Y^n)\leq 1-H(\alpha)$. While the conjecture remains open, we provide substantial evidence supporting its validity.