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

Abstract

The Z p s -additive codes of length n are subgroups of Z n p s , 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 a GH code over Z p which is the image of a 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 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).