$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-Additive Hadamard Codes

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$, and can be seen as linear codes over $\mathbb{Z}_2$ when $\alpha_2=\alpha_3=0$, $\mathbb{Z}_4$-additive or $\mathbb{Z}_8$-additive codes when $\alpha_1=\alpha_3=0$ or $\alpha_1=\alpha_2=0$, respectively, or $\mathbb{Z}_2\mathbb{Z}_4$-additive codes when $\alpha_3=0$. A $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is a Hadamard code which is the Gray map image of a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive code. In this paper, we generalize some known results for $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes to $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, and $\alpha_3 \neq 0$. First, we give a recursive construction of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive Hadamard codes of type $(\alpha_1,\alpha_2, \alpha_3;t_1,t_2, t_3)$ with $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$. Then, we show that in general the $\mathbb{Z}_4$-linear, $\mathbb{Z}_8$-linear and $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard codes are not included in the family of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes with $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, and $\alpha_3 \neq 0$. Actually, we point out that none of these nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^{11}$ is equivalent to a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code of any other type, a $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, or a $\mathbb{Z}_{2^s}$-linear Hadamard code, with $s\geq 2$, of the same length $2^{11}$.