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

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.

Work supported by INDAM. A. Aguglia and V. Siconolfi have been partially supported by the EU under the Italian NRRP of NextGenerationEU, partnership on "Telecommunications of the Future" (PE00000001 - program RESTART, CUP: D93C22000910001) and by the Italian Ministry of University and Research under the Programme "Department of Excellence" Legge 232/2016 (Grant No. CUP - D93C23000100001)

References in corpus (2)

• On the equivalence of certain quasi-Hermitian varieties
• On the geometry of tensor products over finite fields

Cited by in corpus (1)

• On mutually $\mu$-intersecting quasi-Hermitian varieties with some applications