Parameters for sets of type \((m,n)\) in projective planes of prime power order
type: Item
instance of: scholarly article
title: Parameters for sets of type \((m,n)\) in projective planes of prime power order
zbMATH Open document ID: 0935.51005
author: Mauro Biliotti
author: Eliana Francot
published in: Journal of Combinatorial Theory. Series A
publication date: 1999-07-20T00:00:00Z
review text: A set \(\mathcal S\) of \(k\) points in a projective plane of order \(q\) is of type \((m,n)\), \(m < n,\) if each line intersects \(\mathcal S\) in either \(m\) or \(n\) points. The parameters \((m,n,q)\) are called standard if \(q = (n-m)^2\). The authors show that for fixed \(m > 0\) there are only finitely many possible nonstandard parameters with \(q\) a prime power. The proof is achieved by carefully analyzing a diophantine equation, due to \textit{G. Tallini} [J. Geom. 29, 191-199 (1987; Zbl 0618.51004)], which has to be satisfied by the parameters. The authors also devise a method for determining these parameters and they list them all for \(m \leq 1000\). The results are purely arithmetical and the question whether there actually exist sets of type \((m,n)\) with these parameters remains wide open.
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1295476
zbMATH Keywords: sets of type \((m,n)\)
zbMATH Keywords: projective planes of prime power order
zbMATH Keywords: nonstandard parameters
reviewed by: Norbert Knarr
MaRDI profile type: Publication
full work available at URL: jcta.1998.2935
OpenAlex ID: W2091057567
cites work: Q3790208
cites work: Q3851217
cites work: Q5582375
cites work: Sets of type \((m,n)\) in the affine and projective planes of order nine
cites work: Some new results on sets of type (m,n) in projective planes
cites work: Q5524221
DOI: 10.1006/JCTA.1998.2935
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description: scientific article; zbMATH DE number 1295476