The kernel-block rank profile and a complete classification of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

The $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-additive codes are subgroups of $\mathbb{Z}_2^{α_1}\times\mathbb{Z}_4^{α_2}\times\mathbb{Z}_8^{α_3}$, and a $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code is the Gray map image of such a code. A recursive construction of $\mathbb{Z}_2\mathbb{Z}