Some Menon designs having $U(3,3)$ as an automorphism group

Abstract

There exists a unique symmetric (36,15,6) design $\mathcal{D}$ having $G'(2,2) \cong U(3,3)$ as an automorphism group. There is an incidence matrix $M$ of $\mathcal{D}$ which is symmetric with $1$ everywhere on the main diagonal. Thus $\mathcal{D}$ admits a polarity for which all points are absolute. Therefore, $M$ is an adjacency matrix of a strongly regular graph with parameters $(36,14,4,6)$.

Using this design one can produce a series of symmetric designs with parameters $(4 \cdot (3 \cdot 2^k)^2,2 \cdot (3 \cdot 2^k)^2 - 3 \cdot 2^k, (3 \cdot 2^k)^2 - 3 \cdot 2^k)$, $k \in N$, each of which admits an automorphism group isomorphic to the unitary group $U(3,3)$. There is an incidence matrix for each of these designs which is symmetric with constant diagonal. Therefore, these matrices correspond to adjacency matrices of strongly regular graphs.