How many $M\in\{0,1\}^{r\times c}$ are there such that each row and each column of $M$ is distinct?

How many classes of matrices in $\{0,1\}^{r\times c}$ up to permutation equivalence are there such that each matrix in every class has each row and each column distinct?

What is the analogous count if rank over $\Bbb K$ is restricted to exactly $m$ where $\Bbb K$ is a field?

If $\Bbb K=\Bbb R$ then if $r=c$ then asymptotically I think we should have $2^{cm}$ matrices.