Abstract
An orthomorphism over a finite field Fq is a permutation θ:Fq→Fq such that the map x↦θ(x)−x is also a permutation of Fq. The degree of an orthomorphism of Fq, 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∉{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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