Abstract
In this paper we introduce a set of sufficient criteria for the construction of relative hemisystems of the Hermitian space \({\mathrm {H}}(3,q^2)\), unifying all known infinite families. We use these conditions to provide new proofs of the existence of the known infinite families of relative hemisystems. Reproving these results has allowed us to find new relative hemisystems closely related to an infinite family of Cossidente’s, and develop techniques that are likely to be useful in finding relative hemisystems in future.
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Acknowledgments
The authors would like to express their thanks to Prof. Gordon Royle for his assistance in computation for this paper, and Dr. Angela Aguglia for her insight into intersections of hyperbolic quadrics with Hermitian spaces. The first author acknowledges the support of the Australian Research Council Future Fellowship FT120100036. The second author acknowledges the support of a Hackett Postgraduate Research Scholarship. The third author acknowledges the support of the Australian Research Council Discovery Grant DP120101336.
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Communicated by G. Lunardon.
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Bamberg, J., Lee, M. & Swartz, E. A note on relative hemisystems of Hermitian generalised quadrangles. Des. Codes Cryptogr. 81, 131–144 (2016). https://doi.org/10.1007/s10623-015-0135-x
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DOI: https://doi.org/10.1007/s10623-015-0135-x