The structure of elusive codes in Hamming graphs

We consider a code to be a subset of the vertices of a Hamming graph and the set of neighbours are those vertices not in the code, which are distance one from some codeword. An elusive code is a code for which the automorphism group of the set of neighbours is larger than that of the code itself. It is an open question as to whether, for an elusive code, the alphabet size always divides the length of the code. We provide a sufficient condition to ensure that this occurs. Finally, we present a sub-family of the Reed-Muller codes, proving that they are completely transitive and elusive, and that the condition fails for most codes in this sub-family. The length of these examples is again a multiple of the alphabet size.