m-Systems of Polar Spaces

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 ( ν 1 ∪ π 2 ∪ ⋯ ∪ π k ) − π i , i = 1, 2, …, k . In each of the respective cases an upper bound δ for | M | is obtained. 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 . Surprisingly δ is independent of m , giving the explanation why an ovoid and a spread of a polar space P have the same size. In the paper many properties of m -systems are proved. We show that with m -systems of three types of polar spaces there correspond strongly regular graphs and two-weight codes. Also, we describe several ways to construct an m ′-system from a given m -system. Finally, examples of m -systems are given.

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