Is there a good algebraic model of random n-hypergraphs? Suppose $F$ is a finite field and $-1$ is a square in $F$. Let $E$ be the binary relation on $F$ where $(a,b) \in E$ iff $a - b$ is a square. Then $(F,E)$ is called a Paley graph. Paley graphs are well-known algebraic models of random graphs (indeed Paley graphs are ``quasi-random" in a precise sense).
I am curious to know if there is also a "finite field model" for random $n$-hypergraphs. It would be natural to ask for a quasi-random algebraic family of $n$-hypergraphs, but apparently there are multiple notions of quasi-randomness for hypergraphs, and I don't know anything about them, so I don't want to be more precise.
 A: For $G$ a quasirandom hypergraph and $k \geq 3$, let $H$ be a hypergraph whose vertices are that of $G$ and $(x_1, ..., x_k)$ is a hyperedge iff $(x_1, ..., x_k)$ is a clique in $G$. Then there is a one-to-one correspondence between the hyperedges of $H$ and the $k$-cliques of $G$ and between the copies of $M_k$ in $H$ and the copies of $L(Q_k)$ in $G$, where $M_k$, a $k$-hypergraph with $k{2^{k-1}}$ vertices and $2^k$ edges, is defined on p.4 of the paper "Weak quasi-randomness for uniform hypergraphs" and $L(Q_k)$ is the line graph of the hypercube graph.
The $\text{MIN}_d$ condition, a quasi-random condition defined on p.5 of "Weak quasi-randomness for uniform hypergraphs", states that a sequence of hypergraphs is quasirandom if the density of a hyperedge (i.e. probability of finding a hyperedge by choosing $k$ vertices) tends to a fixed $d$ and the density of  $M_k$ tends to $d^{2^k}$.
The density of hyperedges on $H$ tends to $1/2^{k \choose 2}$ and the density of $M_k$ tends to $1/2^{{k \choose 2}2^k}$, by counting $k$-cliques and $L(Q_k)$ in the quasirandom graph $G$. Thus $H$ is quasidrandom if $G$ is.
So we can define such a class of $H$ by taking $G$ to be Paley graphs.
