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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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References:
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