On the Kernel of Z₂ₛ-Linear Hadamard Codes

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The Z₂ₛ-additive codes are subgroups of Zⁿ₂ₛ, and can be seen as a generalization of linear codes over Z₂ and Z₄. A Z₂ₛ-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z₂ₛ-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the Z₄-linear Hadamard codes. In this paper, the kernel of Z₂ₛ-linear Hadamard codes and its dimension are established for s > 2. Moreover, we prove that this invariant only provides a complete classification for some values of t and s. The exact amount of nonequivalent such codes are given up to t=11 for any s≥ 2, by using also the rank and, in some cases, further computations.