M-SYSTEMS OF POLAR SPACES

Let P be a finite classical polar space of rank r, with r greater-than -or-equal-to 2. A partial m-system M of P, with 0 less-than-or-equal-t o m less-than-or-equal-to r - 1, is any set {pi1, pi2,...., pi(k)} of k not-equal 0) totally singular m-spaces of P such that no maximal tot ally singular space containing pi(i) has a point in common with (pi1 o r pi2 or ... or pi(k)) - pi(i), i = 1, 2,..., k. In each of the respec tive cases an upper bound delta for \M\ is obtained. If \M\ = delta, t hen M is called an m-system of P. For m = 0 the m-systems are the ovoi ds of P; for m = r - 1 the m-systems are the spreads of P. Surprisingl y 8 is independent of m, giving the explanation why an ovoid and a spr ead of a polar space P have the same size. In the paper many propertie s of m-systems are proved. We show that with m-systems of three types of polar spaces there correspond strongly regular graphs and two-weigh t codes. Also, we describe several ways to construct an m'-system from a given in-system. Finally, examples of m-systems are given. (C) 1994 Academic Press Inc.