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On the largest caps contained in the Klein quadric of \(PG(5,q)\), \(q\) odd. (English) Zbl 0947.51009
Let \(\mathcal H\) be the Klein quadric of order \(q\), i.e. the hyperbolic quadric in the projective space \(\text{PG}(5,q)\). A \(k\)-cap \(K\) in \(\mathcal H\) is a set of \(k\) points no three of which are collinear. If \(q\) is odd, then a \(k\)-cap \(K\) in \(\mathcal H\) exists only if \(k\leq q^3+q^2+q+1\), and in case of equality \(K\) intersects each plane of \(\mathcal H\) in a conic. Three examples of such caps are known [D. G. Glynn, Geom. Dedicata 26, No. 3, 273-280 (1988; Zbl 0645.51012), A. A. Bruen and J. W. P. Hirschfeld, Eur. J. Comb. 9, No. 3, 255-270 (1988; Zbl 0644.51006)].
The author proves that any \((q^3+q^2+q+1)\)-cap in the Klein quadric \(\mathcal H\), where \(q>3138\) is odd, is the intersection of the Klein quadric with another quadric. Furthermore, he gives restrictions on the type of this quadric. So, he gets close to a classification of caps of maximum size of the Klein quadric.

MSC:
51E22 Linear codes and caps in Galois spaces
05B25 Combinatorial aspects of finite geometries
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