Codes over $\mathbb{Z}_{2^k}$, Gray map and self-dual codes
Steven T. Dougherty, Cristina Fernández-Córdoba · Advances in Mathematics of Communications · 2011
The generalized Gray map is defined for codes over $\mathbb{Z}_{2^k}$. We give bounds for the dimension of the kernel and the rank of the image of a code over $\mathbb{Z}_{2^k}$ with a given type and show that there exists such a code for each dimension in the interval for the kernel. We determine when the Gray image of a code over $\mathbb{Z}_{2^k}$ generates a linear self-dual code and give families of codes whose image generate binary self-dual codes. We investigate the Gray image of quaternary self-dual codes and examine when the Gray image of a self-dual code over $\mathbb{Z}_4$ is a binary self-dual code.