On $Z_pZ_{p^k}$-additive codes and their duality

In this paper, two different Gray-like maps from $Z_p^alphatimes Z_{p^k}^beta$, where $p$ is prime, to $Z_p^n$, $n={alpha+beta p^{k-1}}$, denoted by $phi$ and $Phi$, respectively, are presented. We have determined the connection between the weight enumerators among the image codes under these two mappings. We show that if $C$ is a $Z_p Z_{p^k}$-additive code, and $C^bot$ is its dual, then the weight enumerators of the image $p$-ary codes $phi(C)$ and $Phi(C^bot)$ are formally dual. This is a partial generalization of [On $Z_{2^k}$-dual binary codes, arXiv:math/0509325], and the result is generalized to odd characteristic $p$ and mixed alphabet. Additionally, a construction of $1$-perfect additive codes in the mixed $Z_p Z_{p^2} ... Z_{p^k}$ alphabet is given.