  We consider a code to be a subset of the vertex set of a Hamming graph. We
examine elusive pairs, code-group pairs where the code is not determined by
knowledge of its set of neighbours. We construct a new infinite family of
elusive pairs, where the group in question acts transitively on the set of
neighbours of the code. In our examples, we find that the alphabet size always
divides the length of the code, and prove that there is no elusive pair for the
smallest set of parameters for which this is not the case. We also pose several
questions regarding elusive pairs.
