This question is motivated by the ongoing discussion under my answer to this question. I wrote the following there:

A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called blocks) of an $r$-element set $S$ such that every $p$-element subset of $S$ is contained in a unique element of $A$. Good examples come from considering as blocks the set of hyperplanes in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field. For example, $\mathbb{A}^2$ over $\mathbb{F}_3$ gives a $(2, 3, 9)$ Steiner system: it contains $9$ ($\mathbb{F}_3$-rational) points, and let the blocks be the lines, each of which consists of $3$ points. Then any $2$ points are contained in a unique line. This is the unique $(2, 3, 9)$ Steiner system.

In general, considering $d$-planes in $\mathbb{A}^n$ or $\mathbb{P}^n$ gives an analogous Steiner system, and papers such as this one contain similar constructions.

Loosely, my question is: which Steiner systems come from similar constructions? I'll make this more precise in a bit.

An artificial construction allows us to realize any Steiner system as the points of a variety in $\mathbb{A}^n$ over $\mathbb{F}_2$, the blocks of which are given by the intersection of the variety with some specified hyperplanes, as follows. (This construction is due to Jeremy Booher.) Say we have a $(p,q, r)$ Steiner system with $k$ blocks; consider the subvariety of $\mathbb{A}^k$ containing the point $y_j=(a_i)_{1\leq i\leq k}$ with $a_i=0$ if the $j$-th element in our Steiner system is in block $i$ and $1$ otherwise. Then the intersections with the hyperplanes $x_i=0$ give our blocks. And this subset is a variety as any subset of $\mathbb{A}^k$ is a variety, as it is finite.

So let us try for something harder: Which Steiner systems $(p, q, r)$ come from a subvariety $X$ of $\mathbb{A}^n$ or $\mathbb{P}^n$ containing $r$ ($\mathbb{F}_s$-rational) points, with the blocks given as the intersections with all $p+1$-dimensional hyperplanes? A slightly weaker version: when is there a subvariety $X$ of $\mathbb{A}^n$ or $\mathbb{P}^n$ with $r$ $\mathbb{F}_s$-rational points such that every $p+1$ plane intersects $X$ at $q$ points?

This is probably too hard, so perhaps the simpler question is tractable:

Can you prove that some Steiner system does not come from this construction?