# All $2$-designs arising from the action of the affine linear group on the field of prime order

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.

For fixed $k \ge 3$, the generic such design is a full orbit with "index" $\lambda = k(k-1)$ and is probably not of much (combinatorial) interest. Short orbits are, of course, very interesting. As an example, developing $\{1,2,4\}$ under the field of order $7$ gives the Fano plane. Replace $7$ by a larger prime and this is just a rather boring (setting aside intricate structural questions) design with index six.
• As far as I know for the orbit of a $k$-subset $S$ of $\mathbb{Z}_p$, $\lambda=\frac{k(k-1)}{|Stab_{AL(p)}(S)|}$, where $Stab_{AL(p)}(S)=\{\alpha\in AL(p) \;|\; S^\alpha=S\}$. So you are claiming that $|Stab_{AL(p)}(S)|=1$. Could you please give a hint for this? Nov 8, 2014 at 6:30