, Volume 38, Issue 2, pp 275-284

Relative hemisystems on the Hermitian surface

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Abstract

Let S be a generalized quadrangle of order (q 2,q) containing a subquadrangle S′ of order (q,q). Then any line of S either meets S′ in q+1 points or is disjoint from S′. After Penttila and Williford (J. Comb. Theory, Ser. A 118:502–509, 2011), we call a subset H of the lines disjoint from S′ a relative hemisystem of S with respect to S′, provided that for each point x of SS′, exactly half of the lines through x disjoint from S′ lie in H. A new infinite family of relative hemisystems on the generalized quadrangle \(\mathcal{H}(3,q^{2})\) admitting the linear group PSL(2,q) as an automorphism group is constructed. The association schemes arising from our construction are not equivalent to those arising from the Penttila–Williford relative hemisystems.