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Partitioning projective geometries into caps. (English) Zbl 0571.51002
For q any prime-power and \(n\geq 2\) an even integer, set \(N=(q^ n-1)/(q- 1)\) and \(M=q^ n+1\). A representation of \(\Sigma =PG(2n-1,q)\) in terms of the finite field \(GF(q^{2n})\) is used to partition \(\Sigma\) into N disjoint M-caps. In particular, when \(n=2\) and \(q>2\), a partition of PG(3,q) into \(q+1\) ovoids is obtained. This partition is then used to construct (r,\(\lambda)\)-designs with \(r=(q^ 2+1)/2\) and \(\lambda =(q- 1)^ 2/4\) for every odd prime-power q as well as (v,k,\(\lambda)\)-BIBD’s with \(v=q+1\), \(k=q/2\), and \(\lambda =q(q-2)/4\) for \(q=2^ h\), \(h\geq 2\).

MSC:
51A05 General theory of linear incidence geometry and projective geometries
51E05 General block designs in finite geometry
05A17 Combinatorial aspects of partitions of integers
51E30 Other finite incidence structures (geometric aspects)
05B05 Combinatorial aspects of block designs
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