Hyperplanes of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mi mathvariant="italic">DW</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>2</mml:mn><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si2.gif" overflow="scroll"><mml:mi>q</mml:mi><mml:mo>≠</mml:mo><mml:mn>2</mml:mn></mml:math>, without ovoida

Bruyn, Bart De

doi:10.1016/j.disc.2007.01.014

Discrete Mathematics

2007

DOI: 10.1016/j.disc.2007.01.014

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Hyperplanes of DW(2n-1,q), q≠2, without ovoida

Bart De Bruyn

Abstract: We determine all hyperplanes of DW (2n − 1, q), q = 2, without ovoidal quads. We will show that each such hyperplane either consists of all maximal singular subspaces of W (2n − 1, q) which meet a given (n − 1)-dimensional subspace of PG(2n − 1, q) or (only when q is even) arises from the spin-embedding of DW (2n − 1, q).

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Cited by 3 publications

References 17 publications

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A characterization of a class of hyperplanes of DW(2n−1,F)

Bruyn

2017

Discrete Mathematics

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A hyperplane of the symplectic dual polar space DW (2n−1, F), n ≥ 2, is said to be of subspace-type if it consists of all maximal singular subspaces of W (2n − 1, F) meeting a given (n − 1)-dimensional subspace of PG(2n − 1, F). We show that a hyperplane of DW (2n − 1, F) is of subspace-type if and only if every hex F of DW (2n − 1, F) intersects it in either F , a singular hyperplane of F or the extension of a full subgrid of a quad. In the case F is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane H of DW (2n − 1, F) is of subspace-type or arises from the spin-embedding of DW (2n − 1, F) ∼ = DQ(2n, F) if and only if every hex F intersects it in either F , a singular hyperplane of F , a hexagonal hyperplane of F or the extension of a full subgrid of a quad.

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A characterization of a class of hyperplanes of DW(2n−1,F)

Bruyn

2017

Discrete Mathematics

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On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

Bruyn

2009

Electron. J. Combin.

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Let $\Delta$ be a symplectic dual polar space $DW(2n-1,{\Bbb K})$ or a Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$, $n \geq 2$. We define a class of hyperplanes of $\Delta$ arising from its Grassmann-embedding and discuss several properties of these hyperplanes. The construction of these hyperplanes allows us to prove that there exists an ovoid of the Hermitian dual polar space $DH(2n-1,{\Bbb K},\theta)$ arising from its Grassmann-embedding if and only if there exists an empty $\theta$-Hermitian variety in ${\rm PG}(n-1,{\Bbb K})$. Using this result we are able to give the first examples of ovoids in thick dual polar spaces of rank at least 3 which arise from some projective embedding. These are also the first examples of ovoids in thick dual polar spaces of rank at least 3 for which the construction does not make use of transfinite recursion.

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On Hyperplanes of the Geometry D<sub>4,2</sub> and their Related Codes

Zayda¹

2009

Journal of Mathematics and Statistics

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Problem statement:The point-line geometry of type D 4,2 was introduced and characterized by many authors such as Shult and Buekenhout and in several researches many of geometries were considered to construct good families of codes and this forced us to present very important substructures in such geometry that are hyperplanes. Approach: We used the isomorphic classical polar space Ω + (8, F) and their combinatorics to construct the hyperplanes and the family of certain codes related to such hyperplanes. Results: We proved that each hyperplane is either the set ∆  2 (p) which consisted of all points at a distance mostly 2 from a fixed point p or a Grassmann geometry of type A 3,2 and then we presented a new family of non linear binary constant-weight codes. Conclusion:The hyperplanes of the geometry D 4,2 allow us to discuss further substructures of the geometry such as veldkamp spaces.

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Hyperplanes of DW(2n-1,q), q≠2, without ovoida

Cited by 3 publications

References 17 publications

A characterization of a class of hyperplanes of DW(2n−1,F)

A characterization of a class of hyperplanes of DW(2n−1,F)

On a Class of Hyperplanes of the Symplectic and Hermitian Dual Polar Spaces

On Hyperplanes of the Geometry D<sub>4,2</sub> and their Related Codes

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