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

TL;DR

This paper classifies quasi-Hermitian varieties in even characteristic, showing their projective equivalence, determines their symmetry groups, and constructs new orthogonal arrays for experimental design.

Contribution

It completes the classification of BM quasi-Hermitian varieties in characteristic 2 and explicitly describes their collineation groups, also constructing new orthogonal arrays.

Findings

All quasi-Hermitian varieties in characteristic 2 are projectively equivalent.

The structure of the full collineation group stabilizing these varieties is explicitly determined.

A new family of orthogonal arrays with specific parameters is constructed.

Abstract

In this paper we study the BM quasi-Hermitian varieties introduced in [A. Aguglia, A. Cossidente, G. Korchm\`aros, 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.