On the incompleteness of \((k,n)\)-arcs in Desarguesian planes of order \(q\) where \(n\) divides \(q\)
type: Item
instance of: scholarly article
title: On the incompleteness of \((k,n)\)-arcs in Desarguesian planes of order \(q\) where \(n\) divides \(q\)
zbMATH Open document ID: 0923.51006
DOI: 10.1023/A:1005035602863
author: Simeon Ball
author: Aart Blokhuis
published in: Geometriae Dedicata
publication date: 1999-04-22T00:00:00Z
review text: A \((k, n)\)-arc in a projective plane is a set of \(k\) points, at most \(n\) on every line. A \((k, n)\)-arc is called complete if it cannot be extended to a \((k + 1,n)\)-arc. The authors investigate the completeness of an \((nq - q + n - \varepsilon,n)\)-arc in the Desarguesian plane of order \(q\) where \(n\) divides \(q.\) It is shown that an \((nq - q + n - \varepsilon, n)\)-arc in \(PG(2, q)\), \(n < q/3,\) is incomplete for \(0 < \varepsilon\leqslant n/2\) and any such arc can therefore be completed to a maximal arc. For \(q = 2n\) they are incomplete for \(0 <\varepsilon<0.381n\) and for \(q = 3n\) they are incomplete for \(0 < \varepsilon <0.476n.\) It is known that for \(q\) odd such arcs do not exist for \(\varepsilon = 0\) and, hence, the authors improve the upper bound on the maximum size of such a \((k, n)\)-arc.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B25
zbMATH DE Number: 1270444
zbMATH Keywords: maximal arcs
zbMATH Keywords: \(k\)-arcs
zbMATH Keywords: incompleteness of \((k,n)\)-arcs
zbMATH Keywords: finite Desarguesian planes
zbMATH Keywords: finite geometries
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: a:1005035602863
OpenAlex ID: W1588604260
description: scientific article; zbMATH DE number 1270444