## 51E15 Affine and projective planes

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#### Document Type

- Doctoral Thesis (1)
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#### Keywords

- Differenzmenge (1)
- Endliche Geometrie (1)
- Nicht-Desarguessche Ebene (1)
- Planares Polynom (1)
- Translation planes (1)
- non-desarguesian plane (1)
- ovoids (1)
- planar polynomial (1)
- quadratic forms (1)
- quasiregular group (1)

This thesis discusses methods for the classification of finite projective planes via exhaustive search. In the main part the author classifies all projective planes of order 16 admitting a large quasiregular group of collineations. This is done by a complete search using the computer algebra system GAP. Computational methods for the construction of relative difference sets are discussed. These methods are implemented in a GAP-package, which is available separately. As another result --found in cooperation with U. Dempwolff-- the projective planes defined by planar monomials are classified. Furthermore the full automorphism group of the non-translation planes defined by planar monomials are classified.

In this paper we show that for each prime p=7 there exists a translation plane of order p^2 of Mason-Ostrom type. These planes occur as 6-dimensional ovoids being projections of the 8-dimensional binary ovoids of Conway, Kleidman and Wilson. In order to verify the existence of such projections we prove certain properties of two particular quadratic forms using classical methods form number theory.