Check digit systems over groups and anti-symmetric mappings

A check digit system over a group which detects all single errors and all a djacent transpositions exists if and only if the group possesses an anti-sy mmetric mapping. In this article we give a characterisation for (anti-)auto morphisms to be antisymmetric, show how anti-automorphisms are used to cons truct new anti-symmetric mappings from others and give an upper bound for t he number of anti-symmetric mappings of a group. For groups with sign struc ture, particularly the dihedral group, we present a further construction fo r anti-symmetric mappings. The fact that groups of order 2(2k + 1) have a n on-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 dihedr al group D-m, m odd, no check digit system exists, which detects all jump t ranspositions or all twin errors or all jump twin errors.