Orthomorphisms of groups with minimal possible pairwise distances

Orthomorphisms of groups, which are at the minimum possible distance from each other according to the Cayley metric are studied. A class of transformations is described that map an arbitrary given orthomorphism into the set of all orthomorphisms that are at the minimum possible Cayley distance of two from the original. Using the spectral-difference method for constructing substitutions over the generalized quaternion group $Q_{4 n}$, where $4n = 2^t$ $(t=4,\ldots,8)$, orthomorphisms with values of difference characteristics close to optimal have been found.