On the Courtade-Kumar conjecture for certain classes of Boolean functions

We prove the Courtade-Kumar conjecture, for certain classes of n-dimensional Boolean functions, ∀n ≥ 2 and for all values of the error probability of the binary symmetric channel, ∀0 ≤ p ≤ 1 2 . Let X = [X1 . . . Xn] be a vector of independent and identically distributed Bernoulli( 1 2 ) random variables, which are the input to a memoryless binary symmetric channel, with the error probability in the interval 0 ≤ p ≤ 1 2 , and Y = [Y1 . . . Yn] the corresponding output. Let f : {0, 1}n → {0, 1} be an n-dimensional Boolean function. Then, the Courtade-Kumar conjecture states that the mutual information MI(f(X),Y) ≤ 1−H(p), where H(p) is the binary entropy function.