Check digit systems over groups and anti-symmetric mappings

A check digit system over a group which detects all single errors and all adjacent transpositions exists if and only if the group possesses an anti-symmetric mapping. In this article we give a characterisation for (anti-)automorphisms to be anti-symmetric, show how anti-automorphisms are used to construct new anti-symmetric mappings from others and give an upper bound for the number of anti-symmetric mappings of a group. For groups with sign structure, particularly the dihedral group, we present a further construction for anti-symmetric mappings. The fact that groups of order 2(2k + 1) have a non-trivial sign-structure leads to a very short proof that groups of order 2(2k + 1) possess no complete mapping. Finally we show that over the dihedral group D_m, m odd, no check digit system exists, which detects all jump transpositions or all twin errors or all jump twin errors.