Vol. 1, No. 1, 2005

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Small point sets of $\mathrm{PG}(n,q)$ intersecting each $k$-space in $1$ modulo $\sqrt{q}$ points

Zsuzsa Weiner

Vol. 1 (2005), No. 1, 171–180

The main result of this paper is that point sets in PG(n,q), q = p2h, q 81, p > 2, of size less than 3(qnk + 1)2 and intersecting each k-space in 1 modulo q points (such point sets are always minimal blocking sets with respect to k-spaces) are either (n k)-spaces or certain Baer cones. The latter ones are cones with vertex a t-space, where max{1,n 2k 1} t < n k 1, and with a 2((n k) t 1)-dimensional Baer subgeometry as a base. Bokler showed that non-trivial minimal blocking sets in PG(n,q) with respect to k-spaces and of size at most (qnk+1 1)(q 1)+ q(qnk 1)(q 1) are such Baer cones. The corollary of the main result is that we improve on Bokler’s bound. The improvement depends on the divisors of h; for example, when q is a prime square, we get that the non-trivial minimal blocking sets of PG(n,q) with respect to k-spaces and of size less than 3(qnk + 1)2 are Baer cones.

blocking sets, Baer subgeometries
Mathematical Subject Classification 2000
Primary: 51E20, 51E21
Received: 10 December 2004
Accepted: 21 December 2004
Zsuzsa Weiner