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

Abstract

The ${\mathbb{Z}}_{p^s}$-additive codes of length $n$ are subgroups of ${\mathbb{Z}}_{p^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$, ${\mathbb{Z}}_4$, or ${\mathbb{Z}}_{2^s}$ in general. 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 by a generalized Gray map. A partial classification of these codes by using the dimension of the kernel is known. In this paper, we establish that some ${\mathbb{Z}}_{p^s}$-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 such nonequivalent codes. Moreover, up to $t=10$, this new upper bound coincides with a known lower bound (based on the rank and dimension of the kernel).