On $Z_p Z_{p^k}$ -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 t 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
^⊥
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.