Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitian surface of PG(3, q(2)), q odd; is adopted and a hemisystem on the Hermitian surface H(3, q(2)) admitting the group P Omega(-) (4, q) is constructed, yielding a partial quadrangle PQ((q - 1)/2, q(2), (q - 1)(2)/2) and a strongly regular graph srg((q(3) + 1)(q + 1)/2, (q(2) + 1).(q - 1)/2, (q - 3)/2, (q - 1)(2)/2). For q > 3, no partial quadrangle or strongly regular graph with these parameters was previously known, whereas when q = 3; this is the Gewirtz graph. Thas conjectured that there are no hemisystems on H(3, q(2)) for q > 3, so these are counterexamples to his conjecture. Furthermore, a hemisystem on H(3, 25) admitting 3.A(7).2 is constructed. Finally, special sets (after Shult) and ovoids on H(3, q(2)) are investigated.