An orthomorphism over a finite field $mathbb{F}_q$ is a permutation $theta:mathbb{F}_qmapstomathbb{F}_q$ such that the map $xmapstotheta(x)-x$ 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 $q otin{2, 3, 5, 8}$. 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.
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