Abstract
We first note that each element of a symplectic spread of PG(2n − 1, 2r) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2r). As an application, we show that a translation plane of order q n, q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).
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Bader, L. A remark on symplectic spreads. Ricerche mat. 59, 125–128 (2010). https://doi.org/10.1007/s11587-009-0068-7
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DOI: https://doi.org/10.1007/s11587-009-0068-7