Subgeometries in the André/Bruck–Bose representation

Rottey, Sara; Sheekey, John; Voorde, Geertrui Van de

doi:10.1016/j.ffa.2015.04.002

Cited by 10 publications

(7 citation statements)

References 8 publications

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“…Without loss of generality we may suppose that H = {(1, t, t 2 i ) q k |t ∈ F q k } ∪ {(0, 1, 0) q k , (0, 0, 1) q k } with gcd(i, hk) = 1. The set of affine points of H corresponds to the set of points H = {(1, t, t 2 i ) q ∈ F q ⊕ F q k ⊕ F q k |t ∈ F q k } in PG(2k, q) (for more information about the use of these coordinates for H and H , see [17]). The determined directions in the hyperplane at infinity H ∞ : X 0 = 0 have coordinates (0, t 1 − t 2 , t 2 i 1 − t 2 i 2 ) q where t 1 , t 2 ∈ F q k .…”

Section: Every Translation Hyperoval Defines a Linear Set Of Pseudoregulus Typementioning

confidence: 99%

Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

D'haeseleer¹,

Voorde

2020

Electron. J. Combin.

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In this paper, we study translation hyperovals in PG(2,qk). The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2,qk) are precisely those that have a scattered F2-linear set of pseudoregulus type in PG(2k−1,q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2,q2), see [S.G. Barwick, Wen-Ai Jackson, A characterization of translation ovals in finite even order planes. Finite fields Appl. 33 (2015), 37--52.].

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Section: Every Translation Hyperoval Defines a Linear Set Of Pseudoregulus Typementioning

confidence: 99%

Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

D'haeseleer¹,

Voorde

2020

Electron. J. Combin.

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“…We begin by defining the notion of a scroll that rules two normal rational curves according to a projectivity, see for example [13]. For details on normal rational curves, see [10].…”

Section: Generalising Scrollsmentioning

confidence: 99%

Specialness and the Bose representation

Barwick¹,

Jackson²,

Wild³

2019

Preprint

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This article looks at subconics of order q of PG(2, q 2 ) and characterizes them in the Bruck-Bose representation in PG(4, q). In common with other objects in the Bruck-Bose representation, the characterisation uses the transversals of the regular line spread S associated with the Bruck-Bose representation. By working in the Bose representation of PG(2, q 2 ) in PG(5, q), we give a geometric explanation as to why the transversals of the regular spread S are intrinsic to the characterisation of varieties of PG(2, q 2 ).

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“…Without loss of generality we may suppose that H = {(1, t, t 2 i ) q k |t ∈ F q k } ∪{(0, 1, 0) q k , (0, 0, 1) q k } with gcd(i, hk) = 1. The set of affine points of H corresponds to the set of points H ′ = {(1, t, t 2 i ) q ∈ F q ⊕ F q k ⊕ F q k |t ∈ F q k } in PG(2k, q) (for more information about the use of these coordinates for H and H ′ , see [15]). The determined directions in the hyperplane at infinity H ∞ : X 0 = 0 have coordinates (0, t 1 − t 2 , t 2 i 1 − t 2 i 2 ) q where t 1 , t 2 ∈ F q k .…”

Section: Every Translation Hyperoval Defines a Linear Set Of Pseudore...mentioning

confidence: 99%

Translation hyperovals and $\mathbb{F}_2$-linear sets of pseudoregulus type

D'haeseleer¹,

Voorde²

2019

Preprint

Self Cite

View full text Add to dashboard Cite

In this paper, we study translation hyperovals in PG(2, q k ). The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG(2, q k ) are precisely those that have a scattered F2-linear set of pseudoregulus type in PG(2k − 1, q) as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG(2, q 2 ), see [4].

show abstract

Subgeometries in the André/Bruck–Bose representation

Cited by 10 publications

References 8 publications

Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

Translation Hyperovals and $\mathbb{F}_2$-Linear Sets of Pseudoregulus Type

Specialness and the Bose representation

Translation hyperovals and $\mathbb{F}_2$-linear sets of pseudoregulus type

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