Codes over Z2ᵏ, Gray map and self-dual codes

The generalized Gray map is defined for codes over Z2ᵏ. We give bounds for the dimension of the kernel and the rank of the image of a code over Z2ᵏ 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 Z2ᵏ 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 Z₄ is a binary self-dual code.