Which Boolean Functions Maximize Mutual Information on Noisy Inputs?

We pose a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let X
ⁿ
be independent identically distributed Bernoulli(1/2), and let Yn be the result of passing X
ⁿ
through a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, 1}
ⁿ
→ {0, 1}, we conjecture that I(b(X
ⁿ
); Y
ⁿ
) ≤ 1 - H(α). While the conjecture remains open, we provide substantial evidence supporting its validity. Connections are also made to discrete isoperimetric inequalities.