${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-Additive Generalized Hadamard Codes

Abstract

The ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-additive codes are subgroups of ${\mathbb{Z}}_p^{{\alpha }_1}\times {\mathbb{Z}}_{p^2}^{{\alpha }_2}\times \cdots \times {\mathbb{Z}}_{p^s}^{{\alpha }_s}$, and can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_i=0$ for all $i\in \{2,\dots ,s\}$, a ${\mathbb{Z}}_{p^s}$-additive code when ${\alpha }_i=0$ for all $i\in \{1,\dots ,s-1\}$ , or a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code when $s=2$, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when $p=2$ and $s=2$. A ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}}_p$ which is the Gray map image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-additive code. In this paper, we generalize some known results for ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-linear GH codes with $p$ prime and $s\ge 2$. First, we give a recursive construction of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-additive GH codes of type $({\alpha }_1,\dots ,{\alpha }_s;t_1,\dots ,t_s)$ with $t_1\ge 1,t_2,\dots ,t_{s-1}\ge 0$, and $t_s\ge 1$. Then, we show for which types the corresponding ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}\dots {\mathbb{Z}}_{p^s}$-linear GH codes are nonlinear over ${\mathbb{Z}}_p$. We also compute the kernel and its dimension whenever they are nonlinear.

arXiv admin note: text overlap with arXiv:2203.15657, arXiv:2203.15407