A new hyperoval in \(PG(2,32)\)
type: Item
instance of: scholarly article
title: A new hyperoval in \(PG(2,32)\)
zbMATH Open document ID: 0757.51009
DOI: 10.1007/BF01228288
author: Christine M. O'Keefe
author: Tim Penttila
published in: Journal of Geometry
publication date: 1993-01-16T00:00:00Z
review text: A \(k\)-arc in a projective plane is a set of \(k\) points, no three of which are collinear. An \((n+1)\)-arc in a projective plane of order \(n\) is an oval. When \(n\) is odd, there are no \((n+2)\)-arcs. When \(n\) is odd and the plane is Desarguesian, a well known theorem of Segre says that the oval is a conic. When \(n\) is even, an oval can be extended to a hyperoval i.e., an \((n+2)\)-arc. The authors classify the hyperovals in the Desarguesian plane of order 32 and give one new one. The approach is to represent the hyperovals as sets of points in homogeneous coordinates of the form \(\{(1,t,f(t)\}\cup\{(0,1,0),(0,0,1)\}\) when \(f\) is a permutation polynomial over \(GF(32)\). The author gives the automorphism groups of these hyperovals.
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 51A45
Mathematics Subject Classification ID: 51A20
zbMATH DE Number: 92579
zbMATH Keywords: hyperovals
zbMATH Keywords: Desarguesian plane of order 32
zbMATH Keywords: automorphism groups
reviewed by: Theodore G. Ostrom
MaRDI profile type: Publication
Wikidata QID: Q29395153
cites work: Q3796299
cites work: Q5545191
cites work: Q3312809
cites work: A condition for the existence of ovals in PG(2,q), q even
cites work: Q3851217
cites work: Q3276279
cites work: Q4014431
cites work: Q3769506
cites work: Q3260067
cites work: Q3843291
cites work: Ovali ed altre curve nei piani di Galois di caratteristica due
Recommended article: A new family of dual hyperovals in \(PG(d(d+3)/2,2)\) with \(d\geq 3\)
Recommended article: Classification of hyperovals in \(PG(2,32)\)
Recommended article: Hyperovals in PG(2,16)
Recommended article: New infinite families of hyperovals on \(\mathcal H(3,q^2)\), \(q\) odd
Recommended article: Q4273321
Recommended article: On some \(d\)-dimensional dual hyperovals in \(PG(d(d+3)/2,2\))
Recommended article: Q3973805
Recommended article: On \(d\)-dimensional dual hyperovals in PG\((2d,2)\)
Recommended article: On d-dual hyperovals in
Recommended article: A new \((37, 3)\)-arc in \(\mathrm{PG}(2,23)\)
description: scientific article; zbMATH DE number 92579