Equivalences among Z_p^s-linear Generalized Hadamard Codes
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The _p^s-additive codes of length n are subgroups of _p^s^n, and can be seen as a generalization of linear codes over _2, _4, or _2^s in general. A _p^s-linear generalized Hadamard (GH) code is a GH code over _p which is the image of a _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 _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).
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