Let $p$ be a prime and $\mathbb{Z}_p$ denote as usual the field of order $p$. Let $AL(p)$ be the affine linear group $\{x\mapsto ax+b \;|\; a\in \mathbb{Z}_p\setminus \{0\}, b\in\mathbb{Z}_p\}$. For a given integer $1\leq k \leq p$, it is known that each orbit of the natural action of $AL(p)$ on $k$-subsets of $\mathbb{Z}_p$ is a $2$-design $D_k(p)$.

Are all such $2$-designs $D_k(p)$ known?

Any information on a generic $D_k(p)$ for an arbitrary $p$ and $k$ is apreciated.