An orthomorphism over a finite field ${\mathbb{F}}_q$ is a permutation $\theta :{\mathbb{F}}_q\to {\mathbb{F}}_q$ such that the map $x\mapsto \theta (x)-x$ is also a permutation of ${\mathbb{F}}_q$. The degree of an orthomorphism of ${\mathbb{F}}_q$, that is, the degree of the associated reduced permutation polynomial, is known to be at most $q-3$. We show that this upper bound is achieved for all prime powers $q\notin \{2,3,5,8\}$. We do this by finding two orthomorphisms in each field that differ on only three elements of their domain. Such orthomorphisms can be used to construct 3-homogeneous Latin bitrades.

中文翻译：

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有限域上的正态多项式的次数

有限域上的同构${\mathbb{F}}_q$ 是一个排列 $\theta ：{\mathbb{F}}_q\to {\mathbb{F}}_q$ 使得地图 $X\mapsto \theta (X)-X$ 也是一个排列 ${\mathbb{F}}_q$. 该程度的正形的${\mathbb{F}}_q$，即相关联的约简置换多项式的次数，已知至多为 $q-3$. 我们证明了所有素数都达到了这个上限$q\notin \{2,3,5,8\}$. 我们通过在每个域中找到两个正同态来做到这一点，它们仅在其域的三个元素上有所不同。这种正态可以用来构造 3-homogeneous Latin bitrades。