偶特征下的准埃尔米特簇及相关正交阵列
On quasi-Hermitian varieties in even characteristic and related orthogonal arrays
AI总结：
本文研究特征2、维数3的BM准埃尔米特簇，证明其射影等价性并完成分类，确定其直射变换群结构，还构造了一族元素取自GF(q)的简单正交阵列O(q⁵,q⁴,q,2)。
摘要：
本文研究了特征为2、维数为3的BM准埃尔米特簇，该簇由A. Aguglia、A. Cossidente和G. Korchmàros在《On quasi-Hermitian Varieties》（发表于《J. Combin. Des.》2012年第20期433-447页）中首次提出。在简要探究其组合性质后，我们首先证明所有这类簇在射影意义下都是等价的，这一特性与奇特征下的情况截然不同，相关奇特征研究可参见A. Aguglia和L. Giuzzi发表于《J. Combin. Des.》2022年1-15页的《On the equivalence of certain quasi-Hermitian varieties》。这一结果完成了该论文中启动的分类工作。我们还得到了更进一步的结论：借助已有结果，我们明确确定了稳定这些簇的全部直射变换群的结构。最后，作为本研究的一个副产品，我们还构造了一族简单正交阵列O(q⁵,q⁴,q,2)，其元素取自GF(q)，其中q为偶素数幂。正交阵列（OA）主要用于减少所需的实验次数，以探究测试中变量之间的相互作用。
英文摘要：
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.
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