$\mathbb{Z}_{4}$-linearity cannot be strictly extended to Hamming spaces

The concept of $\mathbb{Z}_{4}$-linearity arises from the labeling of the Hamming Space $(\mathbb{Z}_{2}^{2}, d_{h})$ by the rotation group $\mathbb{Z}_{4}$ and its coordinate-wise extension to $\mathbb{Z}_{2}^{2n}$ \cite{z4}. This labeling establishes a correspondence between several well-known classes of good non-linear binary codes and submodules of $\mathbb{Z}_{4}^{n}$. A natural question should be if $\mathbb{Z}_{4}$-linearity can be extended to other Hamming spaces. A partial and negative answer to this question have been done \cite{ANA}: there is no cyclic labeling of $\mathbb{Z}_{p}^{n}$ for $p$ prime. In this paper we extend this result showing that there is no cyclic labeling for general Hamming spaces. This points out to how special $\mathbb{Z}_{4}$-linearity is and also means that any extension of this concept to Hamming spaces must consider other kinds of labeling groups.