Dictatorship is the Most Informative Balanced Function at the Extremes

Suppose X is a uniformly distributed n-dimensional binary vector and Y is obtained by passing X through a binary symmetric channel with crossover probability α. A recent conjecture by Courtade and Kumar postulates that I(f(X);Y ) ≤ 1− h(α) for any Boolean function f . In this paper, we prove the conjecture for all α ∈ [0, αn], and under the restriction to balanced functions, also for all α ∈ [12−αn, 1 2 ], where αn, αn → 0 as n→∞. In addition, we derive an upper bound on I(f(X);Y ) which holds for all balanced functions, and improves upon the best known bound for all 13 < α < 1 2 .