Let $A$ and $B$ be bit strings (sequences of 0’s and 1’s).
If $A$ is equal to the leftmost length($A$) bits of $B$, then $A$ is said to be a prefix of $B$.
For example, 00110 is a prefix of 001101001, but not of 00111 or 100110.
A prefix-free code of size $n$ is a collection of $n$ distinct bit strings such that no string is a prefix of any other. For example, this is a prefix-free code of size 6:
$$0000, 0001, 001, 01, 10, 11$$
Now suppose that it costs one penny to transmit a ‘0’ bit, but four pence to transmit a ‘1’. Then the total cost of the prefix-free code shown above is 35 pence, which happens to be the cheapest possible for the skewed pricing scheme in question. In short, we write $Cost(6) = 35$.
What is $Cost(10^9)$?