Linear codes of length n over \(\mathbb {Z}_{p^s}\), p prime, called \(\mathbb {Z}_{p^s}\)-additive codes, can be seen as 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 image of a \(\mathbb {Z}_{p^s}\)-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 \(p\ge 3\) prime, we establish that some \(\mathbb {Z}_{p^s}\)-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.

长度为n的线性码在\(\mathbb {Z}_{p^s}\)上，p为素数，称为\(\mathbb {Z}_{p^s}\) -加法码，可以看作子群的\(\mathbb {Z}_{p^s}^n\)。一个\(\mathbb {Z}_{p^s}\) -线性广义 Hadamard (GH) 码是\(\mathbb {Z}_p\)上的 GH 码，它是\(\mathbb { Z}_{p^s}\) -广义灰度图下的加法代码。众所周知，内核的维度允许对这些代码进行部分分类，并建立此类代码数量的一些下限和上限。事实上，在本文中，对于\(p\ge 3\)素数，我们建立了一些\(\mathbb {Z}_{p^s}\)长度为\(p^t\)的线性 GH 代码，具有一旦t固定，相同维度的核就彼此等价。这使我们能够改进已知的上限。此外，直到\(t=10\) if \(p=3\)或\(t=8\) if \(p=5\)，这个新的上限与基于等级的已知下限一致和内核的尺寸。