The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$. 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. 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 $\alpha_1 \neq 0$, $\alpha_2 \neq 0$, $\alpha_3 \neq 0$, $t_1\geq 1$, $t_2 \geq 0$, and $t_3\geq 1$ is known. 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 show for which types the corresponding $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes of length $2^t$ are nonlinear. For these codes, we compute the kernel and its dimension, which allows us to give a partial classification of these codes. Moreover, for $3 \leq t \leq 11$, we give a complete classification by providing the exact amount of nonequivalent such codes. We also prove the existence of several families of infinite such nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, which are not equivalent to any other constructed $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code, nor to any $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code, nor to any previously constructed $\mathbb{Z}_{2^s}$-linear Hadamard code with $s\geq 2$, with the same length $2^t$.

翻译：$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-加性码是$\mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3}$的子群。$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码是$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-加性码的Gray映射像。已知类型为$(\alpha_1,\alpha_2, \alpha_3;t_1,t_2, t_3)$且满足$\alpha_1 \neq 0$、$\alpha_2 \neq 0$、$\alpha_3 \neq 0$、$t_1\geq 1$、$t_2 \geq 0$、$t_3\geq 1$的$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-加性Hadamard码存在递归构造。本文将$\mathbb{Z}_2\mathbb{Z}_4$-线性Hadamard码的若干已知结果推广至满足$\alpha_1 \neq 0$、$\alpha_2 \neq 0$、$\alpha_3 \neq 0$的$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码。首先，我们指出对于哪些类型，相应的长度为$2^t$的$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码是非线性的。对于这些码，我们计算其核及核的维数，从而可对这些码进行部分分类。此外，对于$3 \leq t \leq 11$，我们给出完全分类，精确给出不等价码的数量。我们还证明了若干族无限非线性$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码的存在性，这些码不等价于任何其他已构造的$\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-线性Hadamard码，也不等价于任何$\mathbb{Z}_2\mathbb{Z}_4$-线性Hadamard码，且不等价于任何先前构造的具有相同长度$2^t$且$s\geq 2$的$\mathbb{Z}_{2^s}$-线性Hadamard码。