Let $H$ be an arbitrarily sized matrix drawn randomly and consider a matrix multiplication $Y=HX$ where $X$ can be constructed for the particular $H$ at hand. Partition $Y$ as $$Y=\left[\begin{array}{c}Y_1\\Y_2\\ \vdots \\Y_N\end{array}\right].$$
I want to find the maximum number of columns in a full rank matrix X under constrains $\mathrm{rank}(Y_n)\leq r_n$.
(In my case, the matrices $Y_n$ are of equal size.)
There are many straightforward bounds, for example $X$ can always have at least $\min(\min_n r_n, M)$ columns, where $M$ denotes the number of columns in $H$. However I am interested in exact results.
Further, I am interested in results that hold with probability one when $H$ is randomly drawn, so $H$ can be assumed full rank, etc.