Cones from maximum h-scattered linear sets and a stability result for cylinders from hyperovals

Sam Adriaensen; Jonathan Mannaert; Ferdinando Zullo; Paolo Santonastaso

doi:10.1016/j.disc.2023.113602

Cones from maximum h-scattered linear sets and a stability result for cylinders from hyperovals

Sam Adriaensen, Jonathan Mannaert, Ferdinando Zullo, Paolo Santonastaso

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Abstract

This paper mainly focuses on cones whose basis is a maximum h-scattered linear set. We start by investigating the intersection numbers of such cones with respect to the hyperplanes. Then we analyze two constructions of point sets with few intersection numbers with respect to the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum h-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.

Original language	English
Article number	113602
Number of pages	19
Journal	Discrete Mathematics
Volume	346
Issue number	12
DOIs	https://doi.org/10.1016/j.disc.2023.113602
Publication status	Published - Dec 2023

Bibliographical note

Funding Information:
We would like to thank Jan De Beule, Sam Mattheus and Olga Polverino for fruitful discussions. We are also very grateful to the referees for their suggestions. The third and the fourth authors are very grateful for the hospitality of the Department of Mathematics and Data Science, Vrije Universiteit Brussel, Brussel, Belgium, where the third author was a visiting PhD student for 2 months and the fourth author was a visiting researcher for 1 month during the development of this research. The third and the last authors were supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania Luigi Vanvitelli and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM).

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© 2023 The Author(s)

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2023c - Adriaensen, Mannaert, Santonastaso, Zullo - Cones from maximum h-scattered linear sets and a stabilityFinal published version, 519 KBLicence: CC BY

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abstract = "This paper mainly focuses on cones whose basis is a maximum h-scattered linear set. We start by investigating the intersection numbers of such cones with respect to the hyperplanes. Then we analyze two constructions of point sets with few intersection numbers with respect to the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum h-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.",

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N1 - Funding Information: We would like to thank Jan De Beule, Sam Mattheus and Olga Polverino for fruitful discussions. We are also very grateful to the referees for their suggestions. The third and the fourth authors are very grateful for the hospitality of the Department of Mathematics and Data Science, Vrije Universiteit Brussel, Brussel, Belgium, where the third author was a visiting PhD student for 2 months and the fourth author was a visiting researcher for 1 month during the development of this research. The third and the last authors were supported by the project “VALERE: VAnviteLli pEr la RicErca” of the University of Campania Luigi Vanvitelli and by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA - INdAM). Publisher Copyright: © 2023 The Author(s)

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N2 - This paper mainly focuses on cones whose basis is a maximum h-scattered linear set. We start by investigating the intersection numbers of such cones with respect to the hyperplanes. Then we analyze two constructions of point sets with few intersection numbers with respect to the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum h-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.

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Cones from maximum h-scattered linear sets and a stability result for cylinders from hyperovals

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