Existence results for cyclotomic orthomorphisms

An orthomorphism over a finite field (Formula presented.) is a permutation (Formula presented.) such that the map (Formula presented.) is also a permutation of (Formula presented.). The orthomorphism (Formula presented.) is cyclotomic of index k if (Formula presented.) and (Formula presented.) is constant on the cosets of a subgroup of index k in the multiplicative group (Formula presented.). We say that (Formula presented.) has least indexk if it is cyclotomic of index k and not of any smaller index. We answer an open problem due to Evans by establishing for which pairs (q, k) there exists an orthomorphism over (Formula presented.) that is cyclotomic of least index k. Two orthomorphisms over (Formula presented.) are orthogonal if their difference is a permutation of (Formula presented.). For any list (Formula presented.) of indices we show that if q is large enough then (Formula presented.) has pairwise orthogonal orthomorphisms of least indices (Formula presented.). This provides a partial answer to another open problem due to Evans. For some pairs of small indices we establish exactly which fields have orthogonal orthomorphisms of those indices. We also find the number of linear orthomorphisms that are orthogonal to certain cyclotomic orthomorphisms of higher index.