Propelinear structure of Z_{2k}-linear codes

Let C be an additive subgroup of $Z_{2k}^n$ for any $kgeq 1$. We define a Gray map $Phi:Z_{2k}^n longrightarrow Z_2^{kn}$ such that $Phi(codi)$ is a binary propelinear code and, hence, a Hamming-compatible group code. Moreover, $Phi$ is the unique Gray map such that $Phi(C)$ is Hamming-compatible group code. Using this Gray map we discuss about the nonexistence of 1-perfect binary mixed group code.