In this paper, two different Gray-like maps from Zpα× Zpkβ, where p is prime, to Zpn, n = α t βpk-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 ZpZpk-additive code, and C⊥is its dual, then the weight enumerators of the image p-ary codes φ(C) and Φ(C⊥) are formally dual. This is a partial generalization of [D. S. Krotov, On Z2k-dual binary codes, IEEE Transactions Information Theory 53 (2007), 1532-1537], and the result is generalized to odd characteristic p and mixed alphabet. In addition, a construction of 1-perfect additive codes in the mixed ZpZp2... Zpk alphabet is given.
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Shi et al. (2018) studied this question.