Summary: Let \(G\) be a finite group acting primitively on the sets \(\Omega_1\) and \(\Omega_2\). We describe a construction of 1-designs with block set \(\Omega_1\) and block set \(\Omega_2\), having \(G\) as an automorphism group. Applying this construction method we obtain a unital \(2-(q^3+1, q+1, 1)\), and a semi-symmetric \((q^4-q^3+q^2, q^2-q, (1))\) 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 \(\mathrm{PG}(2, q^2)\). Further, we describe other combinatorial structures constructed from these unitary groups and structures constructed from \(U_4(2), U_4(3)\) and \(L_2(49)\).
We also construct self-orthogonal codes obtained from the row span over \(\mathbb{F}_2\) or \(\mathbb{F}_3\) of the incidence (resp. adjacency) matrices of mostly self-orthogonal designs (resp. strongly regular graphs) defined by the action of the simple unitary groups \(U_3(q)\) for \(q = 3, 4, 7\) and \(U_4(q)\) for \(q = 2, 3\) and the linear group \(L_2(49)\) on the conjugacy classes of some of their maximal subgroups. Some of the codes are optimal or near optimal for the given length and dimension.
For the entire collection see [Zbl 1276.94001].