scholarly journals Calculating the Dimension of the Universal Embedding of the Symplectic Dual Polar Space using Languages

10.37236/9754 ◽  
2020 ◽  
Vol 27 (4) ◽  
Author(s):  
Carlos Segovia ◽  
Monika Winklmeier
Keyword(s):  
Polar Space ◽  
Symplectic Space ◽  
Alternative Proof ◽  
Isotropic Subspace ◽  
Dual Polar Space ◽  
Standard Order ◽  
Isotropic Subspaces ◽  
Universal Embedding ◽  
Totally Isotropic Subspaces ◽  
Symplectic Dual Polar Space

The main result of this paper is the construction of a bijection of the set of words in so-called standard order of length $n$ formed by four different letters and the set $\mathcal{N}^n$ of all subspaces of a fixed $n$-dimensional maximal isotropic subspace of the $2n$-dimensional symplectic space $V$ over $\mathbb{F}_2$ which are not maximal in a certain sense. Since the number of different words in standard order is known, this gives an alternative proof for the formula of the dimension of the universal embedding of a symplectic dual polar space $\mathcal{G}_n$. Along the way, we give formulas for the number of all $n$- and $(n-1)$-dimensional totally isotropic subspaces of $V$.

Bounds on subspace codes based on totally isotropic subspace in symplectic spaces and extended symplectic spaces

2019 ◽  
Vol 12 (05) ◽  
pp. 1950069
Author(s):  
Mahdieh Hakimi Poroch
Keyword(s):  
Sphere Packing ◽  
Symplectic Space ◽  
Isotropic Subspace ◽  
Subspace Codes ◽  
Singleton Bound ◽  
Symplectic Spaces ◽  
Isotropic Subspaces ◽  
Totally Isotropic Subspace ◽  
Totally Isotropic Subspaces

In this paper, we propose the Sphere-packing bound, Singleton bound and Gilbert–Varshamov bound on the subspace codes [Formula: see text] based on totally isotropic subspaces in symplectic space [Formula: see text] and on the subspace codes [Formula: see text] based on totally isotropic subspace in extended symplectic space [Formula: see text].

scholarly journals The universal embedding dimension of the binary symplectic dual polar space

2003 ◽  
Vol 264 (1-3) ◽  
pp. 3-11 ◽  
Author(s):  
A. Blokhuis ◽  
A.E. Brouwer
Keyword(s):  
Polar Space ◽  
Embedding Dimension ◽  
Dual Polar Space ◽  
Universal Embedding ◽  
Symplectic Dual Polar Space

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

10.37236/90 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Hermitian Variety ◽  
Projective Embedding ◽  
Dual Polar Space ◽  
Transfinite Recursion ◽  
Symplectic Dual Polar Space ◽  
Dual Polar Spaces

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.

scholarly journals Non-Classical Hyperplanes of $DW(5,q)$

10.37236/2425 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Bart De Bruyn
Keyword(s):  
Prime Number ◽  
Polar Space ◽  
Dual Polar Space ◽  
Singular Hyperplane ◽  
Symplectic Dual Polar Space

The hyperplanes of the symplectic dual polar space $DW(5,q)$ arising from embedding, the so-called classical hyperplanes of $DW(5,q)$, have been determined earlier in the literature. In the present paper, we classify non-classical hyperplanes of $DW(5,q)$. If $q$ is even, then we prove that every such hyperplane is the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $q$ is odd, then we prove that every non-classical ovoid of $DW(5,q)$ is either a semi-singular hyperplane or the extension of a non-classical ovoid of a quad of $DW(5,q)$. If $DW(5,q)$, $q$ odd, has a semi-singular hyperplane, then $q$ is not a prime number.

Constructions of (r,t)-LRC Based on Totally Isotropic Subspaces in Symplectic Space Over Finite Fields

2020 ◽  
Vol 31 (03) ◽  
pp. 327-339
Author(s):  
Gang Wang ◽  
Min-Yao Niu ◽  
Fang-Wei Fu
Keyword(s):  
Finite Fields ◽  
Linear Codes ◽  
Distributed Storage ◽  
Symplectic Space ◽  
Information Rate ◽  
Local Repair ◽  
Distributed Storage Systems ◽  
Isotropic Subspaces ◽  
Locally Repairable Codes ◽  
Totally Isotropic Subspaces

Linear code with locality [Formula: see text] and availability [Formula: see text] is that the value at each coordinate [Formula: see text] can be recovered from [Formula: see text] disjoint repairable sets each containing at most [Formula: see text] other coordinates. This property is particularly useful for codes in distributed storage systems because it permits local repair of failed nodes and parallel access of hot data. In this paper, two constructions of [Formula: see text]-locally repairable linear codes based on totally isotropic subspaces in symplectic space [Formula: see text] over finite fields [Formula: see text] are presented. Meanwhile, comparisons are made among the [Formula: see text]-locally repairable codes we construct, the direct product code in Refs. [8], [11] and the codes in Ref. [9] about the information rate [Formula: see text] and relative distance [Formula: see text].

scholarly journals On the nucleus of the Grassmann embedding of the symplectic dual polar space DSp(2n,F), char(F)=2

2009 ◽  
Vol 30 (2) ◽  
pp. 468-472 ◽  
Author(s):  
Rieuwert J. Blok ◽  
Ilaria Cardinali ◽  
Bart De Bruyn
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Symplectic Dual Polar Space

scholarly journals The Maximum Size of a Partial Spread in $H(4 n +1, q^2)$ is $q^{2 n +1}+1$

10.37236/251 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Frédéric Vanhove
Keyword(s):  
Maximum Size ◽  
Polar Space ◽  
Maximal Dimension ◽  
Partial Spread ◽  
Isotropic Subspaces ◽  
Hermitian Polar Space ◽  
Totally Isotropic Subspaces

We prove that in every finite Hermitian polar space of odd dimension and even maximal dimension $\rho$ of the totally isotropic subspaces, a partial spread has size at most $q^{\rho+1}+1$, where $GF(q^2)$ is the defining field. This bound is tight and is a generalisation of the result of De Beule and Metsch for the case $\rho=2$.

scholarly journals On the Universal Embedding of the Sp2n(2) Dual Polar Space

2001 ◽  
Vol 94 (1) ◽  
pp. 100-117 ◽  
Author(s):  
Paul Li
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Universal Embedding

scholarly journals Generating Symplectic and Hermitian Dual Polar Spaces over Arbitrary Fields Nonisomorphic to ${\Bbb F}_2$

10.37236/972 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Bart De Bruyn ◽  
Antonio Pasini
Keyword(s):  
Polar Space ◽  
Polar Spaces ◽  
Dual Polar Space ◽  
Symplectic Dual Polar Space ◽  
Dual Polar Spaces

Cooperstein proved that every finite symplectic dual polar space $DW(2n-1,q)$, $q \neq 2$, can be generated by ${2n \choose n} - {2n \choose n-2}$ points and that every finite Hermitian dual polar space $DH(2n-1,q^2)$, $q \neq 2$, can be generated by ${2n \choose n}$ points. In the present paper, we show that these conclusions remain valid for symplectic and Hermitian dual polar spaces over infinite fields. A consequence of this is that every Grassmann-embedding of a symplectic or Hermitian dual polar space is absolutely universal if the (possibly infinite) underlying field has size at least 3.

Two forms related to the symplectic dual polar space in odd characteristic

2011 ◽  
Vol 64 (1-2) ◽  
pp. 47-60 ◽  
Author(s):  
Ilaria Cardinali ◽  
Antonio Pasini
Keyword(s):  
Polar Space ◽  
Dual Polar Space ◽  
Symplectic Dual Polar Space
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