Imagine I have a set of $N$ binary strings of length $L$, where I generate each string randomly (say, by flipping a coin for each bit). What is the probability that all $N$ strings are at least a Hamming distance $k$ apart?
I would be happy with a good lower bound estimate on the probability that all strings are unique. We can estimate the relative sizes of $N$, $L$, and $k$ as: $N >> L$ (by at least an order of magnitude), $5 \leq L \leq 100$, and $k < L$.