Existence results for cyclotomic orthomorphisms
An orthomorphism over a finite field F is a permutation
θ:F→F such that the map x→θ(x)-x is also a
permutation of F. The orthomorphism θ is cyclotomic of
index k if θ(0)=0 and θ(x)/x is constant on the
cosets of a subgroup of index k in the multiplicative group F*.
We say that θ has least index k 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 Fq that is cyclotomic of least index k.
Two orthomorphisms over Fq are orthogonal if their
difference is a permutation of Fq. For any list
[b1,...,bn] of indices we show that if q is
large enough then Fq has pairwise orthogonal orthomorphisms
of least indices b1,...,bn. 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.