On $Z_p{Z}_{p^k}$-additive codes and their duality

Abstract

In this paper, two different Gray-like maps from $Z_p^{\alpha }\times Z_{p^k}^{\beta }$, where $p$ is prime, to $Z_p^n$, $n=\alpha +\beta p^{k-1}$, denoted by $\varphi$ 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 $C^{\perp }$ is its dual, then the weight enumerators of the image $p$-ary codes $\varphi (C)$ and $\Phi (C^{\perp })$ 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.