The kernel-block rank profiles of the $\mathbb{Z}_2\mathbb{Z}_4$-linear and the $\mathbb{Z}_{2^s}$-linear Hadamard codes, and a complete classification of the $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes

The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on