Construction and Linearity of Z_pZ_{p^2}-Linear Generalized Hadamard Codes

Abstract

The ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive codes are subgroups of ${\mathbb{Z}}_p^{{\alpha }_1}\times {\mathbb{Z}}_{p^2}^{{\alpha }_2}$, and can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_2=0$, ${\mathbb{Z}}_{p^2}$-additive codes when ${\alpha }_1=0$, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when $p=2$. A ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear generalized Hadamard (GH) code is a GH code over ${\mathbb{Z}}_p$ which is the Gray map image of a ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive code. In this paper, we generalize some known results for ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with $p=2$ to any $p\ge 3$ prime when ${\alpha }_1\ne 0$. First, we give a recursive construction of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-additive GH codes of type $({\alpha }_1,{\alpha }_2;t_1,t_2)$ with $t_1,t_2\ge 1$. Then, we show for which types the corresponding ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes are non-linear over ${\mathbb{Z}}_p$. Finally, according to some computational results, we see that, unlike ${\mathbb{Z}}_4$-linear GH codes, when $p\ge 3$ prime, the ${\mathbb{Z}}_{p^2}$-linear GH codes are not included in the family of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes with ${\alpha }_1\ne 0$.