Oval designs in Desarguesian projective planes
type: Item
instance of: scholarly article
title: Oval designs in Desarguesian projective planes
zbMATH Open document ID: 0863.51007
DOI: 10.1023/A:1027337923268
author: Laurel L. Carpenter
published in: Designs, Codes and Cryptography
publication date: 1997-06-10T00:00:00Z
review text: From a given hyperoval \({\mathcal O}\) in a projective plane \(\Pi\) of even order \(q\) two designs can easily be obtained. The first is the oval design \({\mathcal W} (\Pi, {\mathcal O})\) which is a resolvable Steiner \(2- ({q \choose 2}, {q \over 2},1)\) design whose points are the exterior lines of \({\mathcal O}\) in \(\Pi\) and whose blocks are the points of \(\Pi\) which are not in \({\mathcal O}\), with incidence inherited from the projective plane. (To set the historical record straight, this construction is correctly attributed to \textit{R. C. Bose} and \textit{M. S. Shrikhande}, but the citation should be to a much earlier paper [Trans. Am. Math. Soc. 95, 191-209 (1960; Zbl 0093.31904)]. In fact, Bose claims to have known of the result since 1949, as reported in a paper by \textit{E. Seiden} [Sankhya, Ser. A 25, 393-394 (1963; Zbl 0124.10803)], who independently rediscovered the construction.) The second design is a Hadamard 3-design, denoted by \({\mathcal H} (\Pi, {\mathcal O})\), with parameters \(3- (q^2, {q^2 \over 2}, {q^2 \over 4} - 1)\) obtained as the unique extension of the Hadamard 2-design whose point set \({\mathcal P}\) is the set of points of \(\Pi\) not on \({\mathcal O}\) and whose blocks are, for each point \(P\in {\mathcal P}\), \(B_P = \{Q\in {\mathcal P} \mid QP\) is a secant of \({\mathcal O}\) in \(\Pi\} \cup \{P\}\). The binary codes of these designs are obtained by taking the block-point incidence matrix of the design as a generator matrix over the binary field, \(F_2\). The dimension of such a code is called its 2-rank. It is known that if the order of the projective plane is \(q= 2^m\), then the 2-rank of the code of the oval design for any hyperoval in that plane is bounded from above by \(3^m-2^m\). The author shows that if the plane is a Desarguesian plane, i.e., \(PG(2,q)\) and the hyperoval is a regular hyperoval (a conic together with its nucleus), then this bound is obtained. It is conjectured that this is true for any…
Mathematics Subject Classification ID: 51E05
Mathematics Subject Classification ID: 05B05
Mathematics Subject Classification ID: 51E21
Mathematics Subject Classification ID: 94B25
zbMATH DE Number: 934917
zbMATH Keywords: Hadamard
zbMATH Keywords: binary codes
zbMATH Keywords: hyperoval
zbMATH Keywords: Reed-Muller code
reviewed by: William E. Cherowitzo
MaRDI profile type: Publication
full work available at URL: a:1027337923268
OpenAlex ID: W1987134520
description: scientific article; zbMATH DE number 934917