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On quasi-Hermitian varieties in even characteristic and related orthogonal arrays

Angela Aguglia, Luca Giuzzi, Alessandro Montinaro, Viola Siconolfi

TL;DR

This work analyzes BM quasi-Hermitian varieties in $PG(3,q^2)$ for even $q$, showing that all such varieties are projectively equivalent and detailing the full stabilizer groups in $PGL_4(q^2)$ and $PΓL_4(q^2)$. It provides a comprehensive description of the geometric structure, including line incidences, and demonstrates transitivity on affine points, culminating in a complete stabilizer description. As a notable application, the authors construct a simple orthogonal array $OA(q^5,q^4,q,2)$ from the BM varieties, linking finite geometry with experimental design. The results extend the understanding of quasi-Hermitian varieties in even characteristic and offer a concrete bridge to OA-based testing designs.

Abstract

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchmàros, On quasi-Hermitian Varieties, J. Combin. Des. 20 (2012) 433-447.] in characteristc $2$ and dimension $3$. After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the classification project started in that paper. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays $O(q^5,q^4,q,2)$, with entries in $\mathrm{GF}{q}$, where $q$ is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed in order to investigate how variables in testing interact with each other.

On quasi-Hermitian varieties in even characteristic and related orthogonal arrays

TL;DR

This work analyzes BM quasi-Hermitian varieties in for even , showing that all such varieties are projectively equivalent and detailing the full stabilizer groups in and . It provides a comprehensive description of the geometric structure, including line incidences, and demonstrates transitivity on affine points, culminating in a complete stabilizer description. As a notable application, the authors construct a simple orthogonal array from the BM varieties, linking finite geometry with experimental design. The results extend the understanding of quasi-Hermitian varieties in even characteristic and offer a concrete bridge to OA-based testing designs.

Abstract

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchmàros, On quasi-Hermitian Varieties, J. Combin. Des. 20 (2012) 433-447.] in characteristc and dimension . After a brief investigation of their combinatorial properties, we first show that all of these varieties are projectively equivalent, exhibiting a behavior which is strikingly different from what happens in odd characteristic, see [A. Aguglia, L. Giuzzi, On the equivalence of certain quasi-Hermitian varieties, J. Combin. Des. 1-15 (2022)]. This completes the classification project started in that paper. Here we prove more; indeed, by using previous results, we explicitly determine the structure of the full collineation group stabilizing these varieties. Finally, as a byproduct of our investigation, we also construct a family of simple orthogonal arrays , with entries in , where is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed in order to investigate how variables in testing interact with each other.
Paper Structure (6 sections, 13 theorems, 128 equations)

This paper contains 6 sections, 13 theorems, 128 equations.

Table of Contents

  1. Introduction
  2. Background on quasi-Hermitian varieties
  3. Preliminaries on BM quasi-Hermitian varieties in $\mathrm{PG}(3,q^2)$, $q$ even
  4. Projective equivalence of $\mathcal{M}_{a,b}$'s
  5. The stabilizer of $\mathcal{M}_{a,b}$
  6. Some orthogonal arrays

Key Result

Theorem 3.1

Let $\mathcal{B}_{a,b}$ be the surface of equation eq:bab in $\mathrm{PG}(3,q^2)$, $q$ an even prime power and put $\mathcal{B}_{\infty}=\mathcal{B}_{a,b}\cap [J=0]$ and $P_{\infty}=(0,0,0,1)$. Then,

Theorems & Definitions (28)

  • Remark 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • proof
  • Remark 3.5
  • Lemma 4.1
  • ...and 18 more