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

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 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 $\alpha_1\neq0$ and $\alpha_2\neq0$, 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 $t_2\geq2$. Finally, we count the nonequivalent codes in the union of both families.