Propelinear structure of Z_{2k}-linear codes

Abstract

Let C be an additive subgroup of ${\mathbb{Z}}_{2k}^n$ for any $k\ge 1$. We define a Gray map $\Phi :{\mathbb{Z}}_{2k}^n⟶{\mathbb{Z}}_2^{kn}$ such that $Φ(\codi)$ 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.