On Quasi-Hermitian Varieties in Even Characteristic and Related Orthogonal Arrays

Abstract

In this article, we study the BM quasi-Hermitian varieties, laying in the three-dimensional Desarguesian projective space of even order. 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. This completes the classification project started there. 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 $O\left(q^5,q^4,q,2\right)$, with entries in $F_q$, where $q$ is an even prime power. Orthogonal arrays (OA's) are principally used to minimize the number of experiments needed to investigate how variables in testing interact with each other.