Abstract
In this paper, a method is developed to study locally hermitian 1-systems of Q(6, q), q even, by associating a kind of flock in PG(4, q) to them. This method is applied to a known locally hermitian 1-system of Q(6, 22e), which was discovered by Offer as a spread of the hexagon H(22e). The results concerning this spread appear to be suitable for generalization and enable us to find new classes of 1-systems of Q(6, q), q even. We also prove that a locally hermitian 1-system of Q(6, q), q even, which is not contained in a 5-dimensional subspace, is semi-classical if and only if it belongs to the new classes we describe. Finally, from the new classes of 1-systems arise new classes of semipartial geometries.
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Luyckx, D., Thas, J.A. On 1-Systems of Q(6, q), q Even. Designs, Codes and Cryptography 29, 179–197 (2003). https://doi.org/10.1023/A:1024112710780
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DOI: https://doi.org/10.1023/A:1024112710780