On Hadamard designs associated with a hyperoval
type: Item
instance of: scholarly article
title: On Hadamard designs associated with a hyperoval
zbMATH Open document ID: 0834.51004
DOI: 10.1007/BF01224045
author: Antonio Maschietti
published in: Journal of Geometry
publication date: 1996-03-18T00:00:00Z
review text: A hyperoval in a projective plane \(\pi\) of order \(q\) is a subset \(\Omega\) of \(q + 2\) points in \(\pi\), no three of which are collinear. Every line of \(\pi\) necessarily meets a hyperoval in zero or two points, the former lines being called exterior lines and the latter ones secant lines. If \(P\) is any point of \(\pi \backslash \Omega\), let \(S(P) = \{Q \in \pi \backslash \Omega : Q \neq P\) and the line \(PQ\) is secant to \(\Omega\} \cup \{P\}\). Suppose \(q = 2^d\), where \(d \geq 3\). Let \(N\) be a point of \(\Omega\), and let \(t\) be a line through \(N\). For every other line \(s\) through \(N\), let \(D_{t,s}\) denote the incident structure whose points are those of \(t \backslash (\Omega \cap t)\) and whose blocks are sets \(S(P) \cap t\) as \(P\) varies over \(s \backslash (s \cap \Omega)\). One can show that \(D_{t,s}\) has \(q-1\) points and blocks, and each block is incident with \({q \over 2} - 1\) points, although \(D_{t,s}\) is not necessarily a 2- design. In the paper under review it is shown that if \(\Omega\) is a translation hyperoval with respect to the line \(t\), then \(D_{t,s}\) is isomorphic to the 2-design of points and hyperplanes of \(PG (d - 1,2)\) for every line \(s\) through \(N,s \neq t\). In particular, \(D_{t,s}\) will be a Hadamard design with parameters \(v = q - 1\), \(k = {q \over 2} - 1\), and \(\lambda = {q \over 4} - 1\).
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 05B30
zbMATH DE Number: 785152
zbMATH Keywords: hyperoval
zbMATH Keywords: Hadamard design
reviewed by: Gary L. Ebert
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 785152