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Designs, Codes and Cryptography

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21-12-2022

Eggs in finite projective spaces and unitals in translation planes

Authors: Giusy Monzillo, Tim Penttila, Alessandro Siciliano

Published in: Designs, Codes and Cryptography | Issue 4/2023

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Abstract

Inspired by the connection between ovoids and unitals arising from the Buekenhout construction in the André/Bruck-Bose representation of translation planes of dimension at most two over their kernel, and since eggs of \(\textrm{PG}(4m-1,q)\), \(m\ge 1\), are a generalization of ovoids, we explore the relation between eggs and unitals in translation planes of higher dimension over their kernel. By investigating such a relationship, we construct a unital in the Dickson semifield plane of order \(3^{10}\), which is represented in \(\textrm{PG}(20,3)\) by a cone whose base is a set of points constructed from the dual of the Penttila-Williams egg in \(\textrm{PG}(19,3)\). This unital is not polar; so, up to the knowledge of the authors, it seems to be a new unital in such a plane.

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Title: Eggs in finite projective spaces and unitals in translation planes
Authors: Giusy Monzillo
Tim Penttila
Alessandro Siciliano
Publication date: 21-12-2022
Publisher: Springer US
Published in: Designs, Codes and Cryptography / Issue 4/2023
Print ISSN: 0925-1022
Electronic ISSN: 1573-7586
DOI: https://doi.org/10.1007/s10623-022-01162-9

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Eggs in finite projective spaces and unitals in translation planes

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