For some $d$-dimensional vector space over a finite field, i.e. $(\mathbb{F}_{p^n})^d$, what is the largest size $s$ so that any set $X\subset (\mathbb{F}_{p^n})^d$ of $s$ distinct points is separated by a rank-$r$ linear map? (i.e. there's some rank-$r$ linear map $T$ where $T(x_1)\neq T(x_2)$ for any distinct $x_1,x_2\in X$.)
Further, having found $s$ for some rank $r$, dimension $d$ over a base field of size $p^n$, then will 'most' linear maps on $\mathbb{F}(p^n)^d$ with rank greater than $r$ separate any $s$ points?
This is equivalent to finding a $(d-r)$-dimensional subspace whose translations all intersect $X$ at at-most one point. Then just choose a map that has that subspace as it's kernel.
I do not work very much with finite fields so I don't know have any non-naive tools to look at this problem with. This is uninteresting in $\mathbb{R}^d$, since there you can find a line (i.e. rank-1 linear map) that separates any finite set of points (project onto any line that isn't in the orthogonal direction of any difference between two points in $X$).