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A common generalization of hypercube partitions and ovoids in polar spaces

Jozefien D'haeseleer, Ferdinand Ihringer, Kai-Uwe Schmidt

Abstract

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural $q$-analog of a subcube partition of the hypercube (which can be seen as a polar space with $q=1$). Our main result proves that a generalized ovoid of $k$-spaces in polar spaces of large rank does not exist. More precisely, for $q=p^h$, $p$ prime, and some positive integer $k$, a generalized ovoid of $k$-spaces in a polar space $\mathcal{P}$ with rank $r \geq r_0(k, p)$ in a vector space $V(n,q)$ does not exist.

A common generalization of hypercube partitions and ovoids in polar spaces

Abstract

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural -analog of a subcube partition of the hypercube (which can be seen as a polar space with ). Our main result proves that a generalized ovoid of -spaces in polar spaces of large rank does not exist. More precisely, for , prime, and some positive integer , a generalized ovoid of -spaces in a polar space with rank in a vector space does not exist.
Paper Structure (13 sections, 14 theorems, 17 equations, 1 table)

This paper contains 13 sections, 14 theorems, 17 equations, 1 table.

Table of Contents

  1. Introduction
  2. Polar Spaces
  3. Generalized Ovoids in Polar Spaces
  4. Homogeneous Generalized Ovoids
  5. Examples for Generalized Ovoids
  6. Non-Homogenous Examples
  7. Recursive construction
  8. Examples of (r, r-1)-ovoids in Q+(2r-1, q), H(2r-1,q), and Q(2r, q)
  9. Examples of (r, r-1)-ovoids in Q+(2r-1, q)
  10. Example of (3, 2)-ovoid in Q+(5, q)
  11. Example of (3, 2)-ovoid in Q-(7, 2)
  12. The Non-Existence of (r, k)-Ovoids for r >> k
  13. Future Work

Key Result

Theorem 1.1

Let $p$ be a prime and let $k$ be a positive integer. Then there exists a constant $r_0(p, k)$ such that for all $r \geq r_0(p, k)$ the following holds: For any positive integer $h$, put $q=p^h$. Let ${\mathcal{P}}$ be a polar space of rank $r$ over the field with $q$ elements. Then ${\mathcal{P}}$

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2: Blokhuis & Moorhouse (1995)
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4: BCN
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • Example 3.4
  • ...and 22 more