I know the $n$-bit message ($M$).
I have to send it to the receiver bit by bit.
For each bit I can also send one bit of comment ($C$).
Before receiver gets the bit, he have to guess it($G$).
After all, $M$, $C$ and $G$ should differ only in $k$ places, ie: $$||(M \underline{\vee} C) \vee (M \underline{\vee} G)|| \leq k $$
Receiver and I both know the protocole of sending the message, so he can guess some informations basing on my comments.
What is the maximal length of the message I can send?
the only solution I've found is $n=2k$, which is pretty obvious (eg. in the first $k$ bits of $C$ I send the second half of the message). Is it possible to construct the protocole in the way, that it allow to send messages longer than $2k$?