Let X be a uniformly distributed binary vector of length n. X is fed into a BSC and produces a random output Y. We seek a function f which maps X to k bits such that the mutual information I(f(X);Y) is maximized. A generalization of a recent conjecture would suggest that this maximum is k C where C is the capacity of the BSC. This bound turns out to be false; an explicit counter example for k=11 can be obtained from the Hamming [15,11,3] code. Moreover, it can be shown that for k and n large enough the maximum mutual information approaches k(1-2p)^2. 

For k=1 the problem remains open but several insights can be obtained. For intance, among functions with a sufficiently large bias lex functions do not maximize mutual information, thereby contradicting another recent conjecture.