On the Rank of Z8-linear Hadamard Codes

© 2018 Elsevier B.V. The Z 2 s -additive codes are subgroups of Z 2 sn , 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≥3 is fixed. In this case, the number of nonequivalent such codes is also established.