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Designs, Codes and Cryptography
Published in: Designs, Codes and Cryptography 2/2016

01-08-2016

Characterising pointsets in \(\mathrm{{PG}}(4,q)\) that correspond to conics

Authors: S. G. Barwick, Wen-Ai Jackson

Published in: Designs, Codes and Cryptography | Issue 2/2016

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Abstract

We consider a non-degenerate conic in \(\mathrm{{PG}}(2,q^2)\), \(q\) odd, that is tangent to \(\ell _\infty \) and look at its structure in the Bruck–Bose representation in \(\mathrm{{PG}}(4,q)\). We determine which combinatorial properties of this set of points in \(\mathrm{{PG}}(4,q)\) are needed to reconstruct the conic in \(\mathrm{{PG}}(2,q^2)\). That is, we define a set \({\mathcal {C}}\) in \(\mathrm{{PG}}(4,q)\) with \(q^2\) points that satisfies certain combinatorial properties. We then show that if \(q\ge 7\), we can use \({\mathcal {C}}\) to construct a regular spread \({\mathcal {S}}\) in the hyperplane at infinity of \(\mathrm{{PG}}(4,q)\), and that \({\mathcal {C}}\) corresponds to a conic in the Desarguesian plane \(\mathcal {P}({\mathcal {S}})\cong \mathrm{{PG}}(2,q^2)\) constructed via the Bruck–Bose correspondence.
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Metadata
Title
Characterising pointsets in that correspond to conics
Authors
S. G. Barwick
Wen-Ai Jackson
Publication date
01-08-2016
Publisher
Springer US
Published in
Designs, Codes and Cryptography / Issue 2/2016
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI
https://doi.org/10.1007/s10623-015-0093-3

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