If you consider all $m$ by $n$ matrices with entries that are either $0$ or $1$, there are ${2^{n} \choose m}$ with no repeated rows (up to row permutation) and ${2^{m} \choose n}$ with no repeated columns (up to column permutation). Is it possible to get an explicit estimate for how many there are with neither any repeated rows nor columns (up to either row or column permutation)? I am interested in the case where $m$ is large and $m \leq n \leq m^2$.

[Extended version of a previous unanswered MSE question ]