Designs from Paley Graphs and Peisert Graphs

Fix positive integers p, q, and r so that p is prime, q = p, and q ≡ 1 (mod 4). Fix a graph G as follows: If r is odd or p 6≡ 3 (mod 4), let G be the qvertex Paley graph; if r is even and p ≡ 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 k4 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 k4. We find estimates and precise asymptotics for k4 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.