Abstract
The concept of a hemisystem of a generalised quadrangle has its roots in the work of B. Segre, and this term is used here to denote a set of points \({\mathcal{H}}\) such that every line ℓ meets \({\mathcal{H}}\) in half of the points of ℓ. If one takes the point-line geometry on the points of the hemisystem, then one obtains a partial quadrangle and hence a strongly regular point graph. The only previously known hemisystems of generalised quadrangles of order (q, q 2) were those of the elliptic quadric \({\mathsf{Q}^-(5,q)}\) , q odd. We show in this paper that there exists a hemisystem of the Fisher–Thas–Walker–Kantor generalised quadrangle of order (5, 52), which leads to a new partial quadrangle. Moreover, we can construct from our hemisystem the 3· A 7-hemisystem of \({\mathsf{Q}^-(5,5)}\) , first constructed by Cossidente and Penttila.
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Communicated by G. Lunardon.
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Bamberg, J., De Clerck, F. & Durante, N. A hemisystem of a nonclassical generalised quadrangle. Des. Codes Cryptogr. 51, 157–165 (2009). https://doi.org/10.1007/s10623-008-9251-1
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DOI: https://doi.org/10.1007/s10623-008-9251-1