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The $\Z_{p^s}$-additive codes of length $n$ are subgroups of $\Z_{p^s}^n$,and can be seen as a generalization of linear codes over $\Z_2$, $\Z_4$, or$\Z_{2^s}$ in general. A $\Z_{p^s}$-linear generalized Hadamard (GH) code is aGH code over $\Z_p$ which is the image of a $\Z_{p^s}$-additive code by ageneralized Gray map. A partial classification of these codes by using thedimension of the kernel is known. In this paper, we establish that some$\Z_{p^s}$-linear GH codes of length $p^t$ are equivalent, once $t$ is fixed.This allows us to improve the known upper bounds for the number of suchnonequivalent codes. Moreover, up to $t=10$, this new upper bound coincideswith a known lower bound (based on the rank and dimension of the kernel).

Equivalences among Z_{p^s}-linear Generalized Hadamard Codes