On $\mathbb{Z}_{\text{8}}$ -Linear Hadamard Codes: Rank and Classification

The Z
_2(s)
-additive codes are subgroups of Z
ⁿ
_2(s)
, and can be seen as a generalization of linear codes over Z
₂
and Z
₄
. A Z
_2(s)
-linear Hadamard code is a binary Hadamard code which is the Gray map image of a Z
_2(s)
-additive code. It is known that either the rank or the dimension of the kernel can be used to give a complete classification for the Z
₄
-linear Hadamard codes. However, when s > 2, the dimension of the kernel of Z
_2(s)
-linear Hadamard codes of length 2
^t
only provides a complete classification for some values of t and s. In this paper, the rank of these codes is computed for s = 3. Moreover, it is proved that this invariant, along with the dimension of the kernel, provides a complete classification, once t ≥ 3 is fixed. In this case, the number of nonequivalent such codes is also established.