Two orthomorphisms over F_{q} are orthogonal if their
difference is a permutation of F_{q}. For any list
[b_{1},...,b_{n}] of indices we show that if q is
large enough then F_{q} has pairwise orthogonal orthomorphisms
of least indices b_{1},...,b_{n}. 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.