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

TL;DR

This paper introduces two Gray-like maps for $Z_p Z_{p^k}$-additive codes, explores their duality properties, and constructs 1-perfect codes in mixed alphabets, extending previous duality results to odd primes.

Contribution

It presents new Gray-like maps for mixed alphabets, analyzes their weight enumerator duality, and constructs 1-perfect additive codes in these settings.

Findings

The weight enumerators of mapped codes are formally dual.

Duality properties extend to odd characteristic primes.

Constructed new 1-perfect additive codes in mixed alphabets.

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 $\phi$ 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 $C^\bot$ is its dual, then the weight enumerators of the image $p$-ary codes $\phi(C)$ and $\Phi(C^\bot)$ 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.