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

In this paper, two different Gray-like maps from $Z_p^α\times Z_{p^k}^β$, where $p$ is prime, to $Z_p^n$, $n={α+βp^{k-1}}$, denoted by $ϕ$ and $Φ$, 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 $ϕ(C)$ and $Φ(C^\bot)$ 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.