Biaffine planes and divisible semiplanes
type: Item
instance of: scholarly article
title: Biaffine planes and divisible semiplanes
zbMATH Open document ID: 0576.51011
DOI: 10.1007/BF01220475
published in: Journal of Geometry
publication date: 1985-01-01T00:00:00Z
review text: 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. \textit{D. R.Hughes} and \textit{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.
Mathematics Subject Classification ID: 51E25
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 3921372
zbMATH Keywords: divisible semibiplanes
zbMATH Keywords: ovals
zbMATH Keywords: conics
author: Peter R. Wild
MaRDI profile type: Publication
cites work: Q5545191
cites work: Q5815962
description: scientific article; zbMATH DE number 3921372