Ovoids in the cyclic presentation of PG(3,q)

Kanat Abdukhalikov; Simeon Ball; Duy Ho; Tabriz Popatia

doi:10.1007/s10623-025-01695-9

Ovoids in the cyclic presentation of PG(3,q)

Kanat Abdukhalikov
, Simeon Ball
, Duy Ho
, Tabriz Popatia

Department of Mathematical Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We consider the cyclic presentation of PG(3,q) whose points are in the finite field Fq4 and describe the known ovoids therein. We revisit the set O, consisting of (q2+1)th roots of unity in Fq4, and prove that it forms an elliptic quadric within the cyclic presentation of PG(3,q). Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of PG(3,q), characterizing them as the zeroes of a polynomial over Fq4.

Original language	English
Pages (from-to)	4765-4778
Number of pages	14
Journal	Designs, Codes, and Cryptography
Volume	93
Issue number	11
DOIs	https://doi.org/10.1007/s10623-025-01695-9
Publication status	Published - Nov 2025

Keywords

Cyclic presentation
Elliptic quadrics
Finite geometries
Ovoids
Projective polynomials
Suzuki–Tits ovoids

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science Applications
Discrete Mathematics and Combinatorics
Applied Mathematics

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10.1007/s10623-025-01695-9

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abstract = "We consider the cyclic presentation of PG(3,q) whose points are in the finite field Fq4 and describe the known ovoids therein. We revisit the set O, consisting of (q2+1)th roots of unity in Fq4, and prove that it forms an elliptic quadric within the cyclic presentation of PG(3,q). Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of PG(3,q), characterizing them as the zeroes of a polynomial over Fq4.",

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AU - Abdukhalikov, Kanat

AU - Ball, Simeon

AU - Ho, Duy

AU - Popatia, Tabriz

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N2 - We consider the cyclic presentation of PG(3,q) whose points are in the finite field Fq4 and describe the known ovoids therein. We revisit the set O, consisting of (q2+1)th roots of unity in Fq4, and prove that it forms an elliptic quadric within the cyclic presentation of PG(3,q). Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of PG(3,q), characterizing them as the zeroes of a polynomial over Fq4.

AB - We consider the cyclic presentation of PG(3,q) whose points are in the finite field Fq4 and describe the known ovoids therein. We revisit the set O, consisting of (q2+1)th roots of unity in Fq4, and prove that it forms an elliptic quadric within the cyclic presentation of PG(3,q). Additionally, following the work of Glauberman on Suzuki groups, we offer a new description of Suzuki–Tits ovoids in the cyclic presentation of PG(3,q), characterizing them as the zeroes of a polynomial over Fq4.

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KW - Elliptic quadrics

KW - Finite geometries

KW - Ovoids

KW - Projective polynomials

KW - Suzuki–Tits ovoids

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Ovoids in the cyclic presentation of PG(3,q)

Abstract

Keywords

ASJC Scopus subject areas

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