On ZpZpk-additive codes and their duality

In this paper, two different Gray-like maps from Z _p ^α × Z _pk ^β , where p is prime, to Z _p ⁿ , n=α +β _p ^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⊥ 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 Z ₂ ^k -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 Z _p Z _p ² ⋯ Z _p ^k alphabet is given.