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 \(\alpha _1 \ne 0\) , \(\alpha _2 \ne 0\) , \(\alpha _3 \ne 0\) , \(t_1\ge 1\) , \(t_2 \ge 0\) , and \(t_3\ge 1\) 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 \(\alpha _1 \ne 0\) , \(\alpha _2 \ne 0\) , and \(\alpha _3 \ne 0\) . 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 \(3 \le t \le 11\) , 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 \(s\ge 2\) , with the same length \(2^t\) .
