Orthomorphisms of Abelian groups with minimum possible pairwise distances

Abstract

Orthomorphisms of Abelian groups with the minimum possible Cayley distance between them are studied. We describe the class of transformations mapping the given orthomorphism into the set of all orthomorphisms such that their Cayley distance from this orthomorphism takes the minimal possible value 2. Algorithms are suggested that construct all orthomorphisms separated from the given one by the minimum possible distance, and the complexity of these algorithms is estimated. Examples of Abelian groups are presented such that the set of all their orthomorphisms contains elements which are separated from all other orthomorphisms by a distance larger than the minimum possible.