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Biaffine planes and divisible semiplanes. (English) Zbl 0576.51011
Let D be a biaffine plane of finite order q, i.e. an affine plane of order q with one parallel class of lines removed. A BAP system in D is a collection C of $$q^ 2$$ ovals in D such that the points of D and the ovals in C are again a biaffine plane. Theorem 3 gives a construction of a (divisible) semibiplane [cf. D. R.Hughes and F. C. Piper, ”Design theory” (1985; Zbl 0561.05009)] from an ”orthogonal” pair of BAP systems in D and its dual. Then BAP systems of conics in desarguesian planes are described in coordinates, and ”orthogonality” is expressed algebraically (Theorem 11). Finally, Theorem 12 shows the existence of (divisible) semibiplanes on $$2q^ 2$$ points with 2q points per block for all prime powers q.
Reviewer: Th.Grundhöfer

##### MSC:
 51E25 Other finite nonlinear geometries 51E20 Combinatorial structures in finite projective spaces 05B25 Combinatorial aspects of finite geometries
##### Keywords:
divisible semibiplanes; ovals; conics
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##### References:
 [1] DEMBOWSKI, P.: Finite Geometrics. Springer-Verlag, Berlin-Heidelberg-New York, 1968. · Zbl 0159.50001 [2] QVIST, B.: Some remarks concerning curves of the second degree in a finite plane. Ann. Acad. Sci. Fenn., no. 134 (1952), p. 1-27. · Zbl 0049.10801 [3] WILD, P.: Ph.D. Thesis, University of London, 1980.
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