On \(d\)-dimensional Buratti-Del Fra type dual hyperovals in \(PG(3d,2)\)
type: Item
instance of: scholarly article
title: On \(d\)-dimensional Buratti-Del Fra type dual hyperovals in \(PG(3d,2)\)
zbMATH Open document ID: 1207.51006
author: Hiroaki Taniguchi
published in: Discrete Mathematics
publication date: 2010-11-19T00:00:00Z
review text: Given an \(m\)-dimensional projective space over \(GF(2)\), \(PG(m, 2)\), and an integer \(d\), where \(m > d \geq 2\), a set \(S\) of \(d\)-dimensional subspaces of \(PG(m, 2)\) is called a \(d\)-dimensional \textit{dual hyperoval} in \(PG(m, 2)\) if: 1) any two distinct elements of \(S\) share a point, 2) any three different elements of \(S\) have a trivial intersection, 3) the union of the elements of \(S\) generates the whole \(PG(m, 2)\) and 4) the cardinality of \(S\) is \(2^{d+1}\); the projective space \(PG(m, 2)\) is called the ambient space of \(S\). Two distinct dual hyperovals \(S_1\) and \(S_2\) that have the same ambient space are isomorphic if there exists a linear automorphism of the ambient space sending the elements of \(S_1\) onto the elements of \(S_2\). For \(d \geq 3\), four non-isomorphic \(d\)-dimensional dual hyperovals are so far known to exist in the ambient space \(PG(d(d+3)/2, 2)\), namely: Huybrechts' dual hyperoval, Buratti-Del Fra's dual hyperoval, Veronesean dual hyperoval and the dual hyperoval that is a deformation of the last-mentioned one. In the present paper, the author -- following and extending the strategy first put forward by S. Yoshiara -- employs a quadratic APN (almost perfect non-linear) function \(f\) on \(GF(2^d)\) and constructs a \(d\)-dimensional dual hyperoval \(D_f\) in \(PG(3d, 2)\) that is a quotient of Buratti-Del Fra's dual hyperoval. Then he proves that given two different quadratic APN functions \(f\) and \(g\) on \(GF(2^d)\), if \(D_f\) is isomorphic to \(D_g\) then \(f\) and \(g\) are extended affine \textit{equivalent} functions on \(GF(2^d)\). The proofs are rather lengthy and entail a lot of technicalities.
reviewed by: Metod Saniga
Mathematics Subject Classification ID: 51E21
zbMATH DE Number: 5817556
zbMATH Keywords: dual hyperoval
zbMATH Keywords: quadratic APN function
zbMATH Keywords: APN (almost perfect non-linear)
MaRDI profile type: Publication
full work available at URL: j.disc.2010.09.014
OpenAlex ID: W2151704972
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DOI: 10.1016/J.DISC.2010.09.014
description: scientific article; zbMATH DE number 5817556