Unitals, Projective Planes and Other Combinatorial Structures Constructed from the Unitary Groups $U(3,q),q=3,4,5,7$

Abstract

Let $G$ be a finite permutation group acting primitively on sets ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$. We describe a construction of a $1$-design
with the block set $\mathcal{B}$ and the point set ${\mathrm{\Omega }}_2$, having $G$ as an automorphism group.Applying this method, we construct a unital $2$-$(q^3+1,q+1,1)$ design and a semi-symmetric design $(q^4-q^3+q^2,q^2-q,(1))$ from the unitary group $U(3,q)$, where $q=3,4,5,7$.From the unital and the semi-symmetric design, we build a projective plane $PG(2,q^2)$. Further, we describe other combinatorial structures constructed from these unitary groups.