On the Rank of Z8-linear Hadamard Codes

The $Z_{2^s}$-additive codes are subgroups of $Z_{2^s}^n$, 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 given for $s=3$. Moreover, it is shown that this invariant, along with the dimension of the kernel, provides a complete classification, once $t\ge 3$ is fixed. In this case, the number of nonequivalent such codes is also established.