Designs from Paley graphs and Peisert graphs

Fix positive integers $p,q,$ and $r$ so that $p$ is prime, $q=p^r$, and $q\equiv 1$ (mod $4$). Fix a graph $G$ as follows: If $r$ is odd or $p\not\equiv 3$ (mod $4$), let $G$ be the $q$-vertex Paley graph; if $r$ is even and $p\equiv 3$ (mod $4$), let $G$ be either the $q$-vertex Paley graph or the $q$-vertex Peisert graph. We use the subgraph structure of $G$ to construct four sequences of $2$-designs, and we compute their parameters. Letting $k_4$ denote the number of $4$-vertex cliques in $G$, we create $62$ additional sequences of $2$-designs from $G$, and show how to express their parameters in terms of only $q$ and $k_4$. We find estimates and precise asymptotics for $k_4$ in the case that $G$ is a Paley graph. We also explain how the presented techniques can be used to find many additional $2$-designs in $G$. All constructed designs contain no repeated blocks.