Some non-existence results on m-ovoids in classical polar spaces

Jan De Beule; Jonathan Mannaert; Valentino Smaldore

doi:10.1016/j.ejc.2024.103943

Some non-existence results on m-ovoids in classical polar spaces

Jan De Beule, Jonathan Mannaert, Valentino Smaldore

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Abstract

In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces
$\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$ for $r>2$. In Bamberg et al. (2009) a lower bound on
$m$ for the existence of $m$-ovoids of $\h(4,q^2)$ is found by using
the connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approach
is generalized in Bamberg et al. (2012) for the polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$, $r>2$.
In \cite{BDS} an improvement for the particular case $\h(4,q^2)$ is obtained by exploiting the algebraic
structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set.
In this paper, we use an approach based on geometrical and combinatorial arguments, inspired by
the results from Gavrilyuk et al. (2023), to improve the bounds from Bamberg et al. (2007) .

Original language	English
Article number	103943
Number of pages	14
Journal	European Journal of Combinatorics
Volume	118
DOIs	https://doi.org/10.1016/j.ejc.2024.103943
Publication status	Published - May 2024

Bibliographical note

Funding Information:
The authors thank the anonymous referees for their careful reading and valuable comments and Francesco Pavese for the fruitful discussions.

Publisher Copyright:
© 2024 Elsevier Ltd

Keywords

m-Ovoids
ovoids
Polar space

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DBMS
I wish to use the Belgian Federal Law of 2018, more particularly:, Art. XI.196, §2/1 van het Wetboek van Economisch Recht, referring to the following article, which gives me the right to make publicly available the manuscript of the publication six months after the original publication appeared. Note that the online publication date of this paper is March 1st, 2024.
Accepted author manuscript, 561 KBLicence: CC BY-NC-ND

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title = "Some non-existence results on m-ovoids in classical polar spaces",

abstract = "In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$ for $r>2$. In Bamberg et al. (2009) a lower bound on $m$ for the existence of $m$-ovoids of $\h(4,q^2)$ is found by usingthe connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approachis generalized in Bamberg et al. (2012) for the polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$, $r>2$.In \cite{BDS} an improvement for the particular case $\h(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired bythe results from Gavrilyuk et al. (2023), to improve the bounds from Bamberg et al. (2007) . ",

keywords = "m-Ovoids, ovoids, Polar space",

author = "{De Beule}, Jan and Jonathan Mannaert and Valentino Smaldore",

note = "Funding Information: The authors thank the anonymous referees for their careful reading and valuable comments and Francesco Pavese for the fruitful discussions. Publisher Copyright: {\textcopyright} 2024 Elsevier Ltd",

year = "2024",

month = may,

doi = "10.1016/j.ejc.2024.103943",

language = "English",

volume = "118",

journal = "European Journal of Combinatorics",

issn = "0195-6698",

publisher = "Academic Press Inc.",

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T1 - Some non-existence results on m-ovoids in classical polar spaces

AU - De Beule, Jan

AU - Mannaert, Jonathan

AU - Smaldore, Valentino

N1 - Funding Information: The authors thank the anonymous referees for their careful reading and valuable comments and Francesco Pavese for the fruitful discussions. Publisher Copyright: © 2024 Elsevier Ltd

PY - 2024/5

Y1 - 2024/5

N2 - In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$ for $r>2$. In Bamberg et al. (2009) a lower bound on $m$ for the existence of $m$-ovoids of $\h(4,q^2)$ is found by usingthe connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approachis generalized in Bamberg et al. (2012) for the polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$, $r>2$.In \cite{BDS} an improvement for the particular case $\h(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired bythe results from Gavrilyuk et al. (2023), to improve the bounds from Bamberg et al. (2007) .

AB - In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$ for $r>2$. In Bamberg et al. (2009) a lower bound on $m$ for the existence of $m$-ovoids of $\h(4,q^2)$ is found by usingthe connection between $m$-ovoids, two-character sets, and strongly regular graphs. This approachis generalized in Bamberg et al. (2012) for the polar spaces $\q^-(2r+1,q), \w(2r-1,q)$ and $\h(2r,q^2)$, $r>2$.In \cite{BDS} an improvement for the particular case $\h(4,q^2)$ is obtained by exploiting the algebraic structure of the collinearity graph, and using the characterization of an $m$-ovoid as an intruiging set. In this paper, we use an approach based on geometrical and combinatorial arguments, inspired bythe results from Gavrilyuk et al. (2023), to improve the bounds from Bamberg et al. (2007) .

KW - m-Ovoids

KW - ovoids

KW - Polar space

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U2 - 10.1016/j.ejc.2024.103943

DO - 10.1016/j.ejc.2024.103943

M3 - Article

SN - 0195-6698

VL - 118

JO - European Journal of Combinatorics

JF - European Journal of Combinatorics

M1 - 103943

ER -

Some non-existence results on m-ovoids in classical polar spaces

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Keywords

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Bounds for m-ovoids using combinatorial techniques

Some non-existence results on m-ovoids in finite classical polar spaces

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Some non-existence results on m-ovoids in classical polar spaces

Abstract

Bibliographical note

Keywords

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Bounds for m-ovoids using combinatorial techniques

Some non-existence results on m-ovoids in finite classical polar spaces

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