Classification of the Z2Z4-linear Hadamard codes and their automorphism groups

A $Z_2Z_4$-linear Hadamard code of length $\alpha+2\beta=2^t$ is a binary Hadamard code which is the Gray map image of a $Z_2Z_4$-additive code with $\alpha$ binary coordinates and $\beta$ quaternary coordinates. It is known that there are exactly $[(t-1)/2]$ and $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$, with $\alpha=0$ and $\alpha\not=0$, respectively, for all $t\geq 3$. In this paper, it is shown that each $Z_2Z_4$-linear Hadamard code with $\alpha=0$ is equivalent to a $Z_2Z_4$-linear Hadamard code with $\alpha\not=0$; so there are only $[t/2]$ nonequivalent $Z_2Z_4$-linear Hadamard codes of length $2^t$. Moreover, the order of the monomial automorphism group for the $Z_2Z_4$-additive Hadamard codes and the permutation automorphism group of the corresponding $Z_2Z_4$-linear Hadamard codes are given.