Luyckx, Deirdre

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Ghent University Academic Bibliography

New 1-systems of Q(6, q), q evenDeirdre Luyckx 1Joint work with J. A. ThasGhent University, Dept. of Pure Mathematics and Computer AlgebraGalglaan 2, B–9000 Ghent, Belgiumdluyckx@cage.rug.ac.beA 1-systemM of the parabolic quadricQ(6, q) in PG(6, q) is a set {L0, L1, . . . ,Lq3} consisting of q3 + 1 lines on Q(6, q) having the property that the tangentspace of Q(6, q) at Li has no point in common with (L0 ∪ L1 ∪ . . . ∪ Lq3) \ Li,i = 0, 1, . . . , q3. We will discuss a method to construct new locally hermitian1-systems of Q(6, q), q even; for q odd, this was already done in [1]. One of these1-systems is the spread of the hexagon H(q), q = 22e, which was discovered byA. Offer in [3]. Moreover, we can classify these new 1-systems as the only oneson Q(6, q) which are locally hermitian and semiclassical, but not contained in a5-dimensional subspace.Our class of new 1-systems has beautiful applications in a wide range offields. By projection from the nucleus of Q(6, q) onto a PG(5, q) not containingthe nucleus, every 1-system of Q(6, q), q even, yields a 1-system of W5(q), hencewe have also found a new class of 1-systems of W5(q). In [2], it is explainedthat every 1-system of W5(q) yields a semipartial geometry, while by a corollaryin [4], a 1-system of W5(q) defines a strongly regular graph and a two-weightcode. So our new class of 1-systems provides us with new examples of semipartialgeometries, strongly regular graphs and two-weight codes.References[1] D. Luyckx and J. A. Thas, Flocks and locally hermitian 1-systems ofQ(6, q),In Finite Geometries - Proceedings of the Fourth Isle of Thorns Conference,volume 3 of Developments in Mathematics, 2001, 257 –275.[2] D. Luyckx, m-systems of polar spaces and SPG reguli, Bulletin of the Bel-gian Mathematical Society – Simon Stevin, 2001, to appear.[3] A. Offer, Spreads and Ovoids of the Split Cayley Hexagon,University of Adelaide, 2000, second edition, available athttp://cage.rug.ac.be/˜ nick/Theses/theses.html[4] E.E. Shult and J.A. Thas, m-systems of polar spaces, J. Combin. TheorySer. A 68 (1994), 184–204.1The author is Research Assistant of the Fund for Scientific Research – Flanders (Belgium)(F.W.O.)

New 1-systems of Q(6,q), q even

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New 1-systems of Q(6,q), q even

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Ghent University Academic Bibliography