Nonequivalence of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear and ${\mathbb{Z}}_{p^s}$-linear generalized Hadamard codes for odd primes

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 they can be seen as linear codes over ${\mathbb{Z}}_p$ when ${\alpha }_2=0$, as ${\mathbb{Z}}_{p^2}$-additive codes when ${\alpha }_1=0$, and as ${\mathbb{Z}}_2{\mathbb{Z}}_4$-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. For every prime $p$, there are recursive constructions of ${\mathbb{Z}}_p{\mathbb{Z}}_{p^2}$-linear GH codes $H_p^{t_1,t_2}$ with ${\alpha }_1\mathrm{\ne }0$ and ${\alpha }_2\mathrm{\ne }0$, and of ${\mathbb{Z}}_{p^s}$-linear GH codes ${\bar{H}}_p^{t_1,\dots ,t_s}$. We show that the kernel blocks of a nonlinear code $H_p^{t_1,t_2}$ give equivalent punctured codes, and that these punctured codes are linear. Any of them is called the derived code of $H_p^{t_1,t_2}$. We also determine the ranks of the codes $H_p^{t_1,t_2}$ and ${\bar{H}}_p^{t_1,t_2}$. For odd $p$, the rank of $H_p^{t_1,t_2}$ depends on the Bernoulli numbers modulo $p$. By comparing the derived codes and the ranks, we show that, for odd $p$, no code $H_p^{t_1,t_2}$ is equivalent to any of these ${\mathbb{Z}}_{p^s}$-linear GH codes. For $p=2$, a nonlinear code $H_2^{t_1,t_2}$ is equivalent to a ${\mathbb{Z}}_{2^s}$-linear Hadamard code if and only if $t_2\ge 2$. Finally, we count the nonequivalent codes in the union of both families.