Small complete caps in PG\((2,q)\), for \(q\) an odd square
type: Item
instance of: scholarly article
title: Small complete caps in PG\((2,q)\), for \(q\) an odd square
zbMATH Open document ID: 0969.51012
DOI: 10.1007/BF01237479
published in: Journal of Geometry
publication date: 2001-03-20T00:00:00Z
review text: A cap \(K\) in a projective space PG\((d,q)\) of dimension \(d\) over the finite field GF\((q)\) is a set of points no three of which are collinear. A cap \(K\) is complete if any point outside \(K\) is a bisecant of \(K\). A cap having \(n\) points is called an \(n\)-cap. For \(d=2\) caps are also called arcs. By using a set of two conics the author constructs a class of \(4(\sqrt q - 1)\)-arcs in PG\((2,q)\) for \(q=m^2\), \(m = p^h\), \(p\) an odd prime. By using the computer completeness and uncompleteness of several of these arcs are proved. Based on this data the main theorem states that for any odd square \(q\) with \(q \leq 1681\) or \(q = 2401\) there exists a complete \(4(\sqrt q - 1)\)-arc in PG\((2,q)\).
Mathematics Subject Classification ID: 51E20
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 1565897
zbMATH Keywords: projective plane
zbMATH Keywords: complete arc
zbMATH Keywords: complete cap
zbMATH Keywords: collineation
author: Massimo Giulietti
MaRDI profile type: Publication
cites work: Q3851217
cites work: Q4273314
description: scientific article; zbMATH DE number 1565897