Labelings of Lee and Hamming spaces

AbstractThe labeling of the Hamming Space (Z22,dh) by the rotation group Z4 and its coordinate-wise extension to Z22n give rise to the concept of Z4-linearity. Attempts to extend this concept have been done in different ways. We deal with a natural extension question: Is there any pattern of a cyclic group G labeling of Zmn with the Hamming or Lee metric? The answer is no. Actually, we show here that Lee spaces do not allow even labelings by abelian groups, what lead us to construct labelings by semi-direct products of abelian groups. Labelings of general Hamming spaces and of Reed–Muller codes RM(1,m) are characterized here in the context of isometry groups