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 \(\Omega_1\) and \(\Omega_2\). We describe a construction of a \(1\)-design
with the block set \(\mathcal{B}\) and the point set \(\Omega_2\), 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.