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From semifield flocks to the generalized translation dual of a semifield - MaRDI portal

From semifield flocks to the generalized translation dual of a semifield (Q1700421)

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scientific article; zbMATH DE number 6847113
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From semifield flocks to the generalized translation dual of a semifield
scientific article; zbMATH DE number 6847113

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    From semifield flocks to the generalized translation dual of a semifield (English)
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    5 March 2018
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    \textit{J. A. Thas} [J. Comb. Theory, Ser. A 79, No. 2, 223--254 (1997; Zbl 0887.51004)] developed relationships between translation generalized quadrangles and symplectic semifield spreads. Sometimes, there are interesting and important links between geometrical objects. For instance, \textit{J. A. Thas} [Eur. J. Comb. 8, 441--452 (1987; Zbl 0646.51019)] proved that given a flock \({\mathcal F}\) of a quadratic cone of \(\mathrm{PG}(3,q)\), then there can be a generalized quadrangle \(Q({\mathcal F})\) associated to this flock \({\mathcal F}\). This means that new results on one of the geometrical objects can lead to new results on the other geometrical objects. This article discusses these relationships between translation generalized quadrangles and symplectic semifield spreads, and discusses some open problems. The discussed topics include: translation ovoids of \(Q(4,q)\) and semifield flocks, translation ovoids of the Klein quadric \(Q^+(5,q)\), geometric spread sets and semifields, and generalized translation dual of a semifield.
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    ovoids
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    semifield flocks
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    translation generalized quadrangles
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    symplectic semifield spreads
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    geometric spread sets
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    polar spaces
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