Full classification of $\mathbb{Z}_{p^s}$-linear generalized Hadamard codes

The $\mathbb{Z}_{p^s}$-additive codes are subgroups of $\mathbb{Z}_{p^s}^n$. A $\mathbb{Z}_{p^s}$-linear generalized Hadamard (GH) code is a GH code over $\mathbb{Z}_p$ which is the Gray map image of a $\mathbb{Z}_{p^s}$-additive code. A recursive construction gives such codes $H_p^{t_1,\dots,t_s}$,