On the equivalence of ℤ_p^s -linear generalized Hadamard codes

Linear codes of length n over ℤ_p^s , p prime, called ℤ_p^s -additive codes, can be seen as subgroups of ℤ_p^s^n . A ℤ_p^s -linear generalized Hadamard (GH) code is a GH code over ℤ_p which is the image of a ℤ_p^s -additive code under a generalized Gray map. It is known that the dimension of the kernel allows to classify these codes partially and to establish some lower and upper bounds on the number of such codes. Indeed, in this paper, for p≥ 3 prime, we establish that some ℤ_p^s -linear GH codes of length p^t having the same dimension of the kernel are equivalent to each other, once t is fixed. This allows us to improve the known upper bounds. Moreover, up to t=10 if p=3 or t=8 if p=5 , this new upper bound coincides with a known lower bound based on the rank and dimension of the kernel.