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Unitals in the Hölz design on 28 points. (English) Zbl 0731.05010

The Hölz design of order 3 is a \(2\)-\((28,4,5)\) design constructed from the point set of a Hermitian unital in \(\mathrm{PG}(2,9)\) by adding new blocks (the intersections of the unital with certain Baer subplanes) to the blocks of the unital. Besides containing the initial Hermitian unital, the design also contains unitals of Ree type as subdesigns. In this paper it is shown that any unital (a \(2\)-\((28,4,1)\) design in the present situation) which occurs as a subdesign must be of either the Hermitian or Ree type. The author proceeds by counting all the unitals of the given types and then verifying by computer that no additional unitals exist.

MSC:

05B30 Other designs, configurations
51E15 Finite affine and projective planes (geometric aspects)
51E30 Other finite incidence structures (geometric aspects)
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