Let G be a finite group acting primitively on the sets Ω1 and Ω2. We describe a construction of 1-designs with block set Ω1 and block set Ω2, having G as an automorphism group. Applying this construction method we obtain a unital 2-(q3+1, q+1, 1), and a semi-symmetric (q4−q3+q2, q2−q, (1)) from the unitary group U3(q), where q = 3, 4, 5, 7. From the unital and the semi-symmetric design we build a projective plane PG(2, q2). Further, we describe other combinatorial structures constructed from these unitary groups and structures constructed from U4(2), U4(3) and L2(49).
We also construct self-orthogonal codes obtained from the row span over 2 or 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 U3(q) for q = 3, 4, 7 and U4(q) for q = 2, 3 and the linear group L2(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.
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