For a fixed ground set $[n]=\{1,\ldots,n\}$, and for any matroid $M$ on $[n]$, specified as a collection of bases $B_M$, the corresponding matroid basis polytope $P_M$ is defined to be the convex hull of the indicator vectors of elements in $B_M$. It follows easily that each such indicator vector will be a vertex of $P_M$, and all the indicator vectors have the same Hamming weight (number of ones), which is equal to the rank of the matroid $M$.
I am interested in matroid constructions that could be associated in some structured way to polytopes, whose vertices are $\{0,1\}$-vectors with the same Hamming weight, but which need not be matroid basis polytopes themselves. In other words, the question is the following: consider a $k$-uniform hypergraph $H$ on the vertex set $[n]$. Is there a matroid that could be associated to $H$ in some structured sense?
My motivation for this is one such construction by Amini and Branden (Theorem 6.3 in https://arxiv.org/abs/1512.05878) which turns out to always be a sparse paving matroid, given as follows: Consider a ground set $G_{2n}$ of $2n$ elements, labeled as $G_{2n}=\{1,\ldots,n\}\sqcup\{1',\ldots,n'\}$. For a $k$-uniform hypergraph $H$ on vertex set $[n]$, denote $A'=\{i':i\in A\}$ for each $A\in E(H)$. The collection $$B(H)=\binom{G_{2n}}{2k}\setminus\{A\sqcup A':A\in E(H)\}$$ is the collection of bases of a sparse paving matroid.
I would think that the above association of a matroid defined by the collection of bases $B(H)$ (on a ground set of $2n$ elements) to any $k$-uniform hypergraph $H$ (on a vertex set of $n$ elements) is structured, in that each such matroid is sparse paving. Are there other such structured matroid constructions known, associated to uniform hypergraphs?
