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m-Systems and Partial m-Systems of Polar Spaces

In: Designs and Finite Geometries

Author

Listed:
  • E. E. Shult

    (Kansas State University, Department of Mathematics)

  • J. A. Thas

    (University of Ghent, Department of Pure Mathematics and Computer Algebra)

Abstract

Let P be a finite classical polar space of rank r, with r ≥ 2. A partial m-system M of P, with 0 ≤ m ≤ r - 1, is any set (π 1, π 2,..., π k } of k (≠ 0) totally singular m-spaces of P such that no maximal totally singular space containing π i has a point in common with (π 1 ∪ π 2 ∪... ∪π k ) - π i , i = 1, 2,..., k. In a previous paper an upper bound δ for ∣M∣ was obtained (Theorem 1). If ∣M∣ = δ, then M is called an m-system of P. For m = 0 the m-systems are the ovoids of P; for m = r - 1 the m-systems are the spreads of P. In this paper we improve in many cases the upper bound for the number of elements of a partial m-system, thus proving the nonexistence of several classes of m-systems.

Suggested Citation

  • E. E. Shult & J. A. Thas, 1996. "m-Systems and Partial m-Systems of Polar Spaces," Springer Books, in: Dieter Jungnickel (ed.), Designs and Finite Geometries, pages 229-238, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-1395-3_17
    DOI: 10.1007/978-1-4613-1395-3_17
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