Z2Z4-Linear Hadamard Codes and Their Automorphism Groups

A Z2Z4-linear Hadamard code of length α+2β = 2 is a binary Hadamard code which is the Gray map image of a Z2Z4-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly b t−1 2 c and b t 2 c nonequivalent Z2Z4-linear Hadamard codes of length 2, with α = 0 and α 6= 0, respectively, for all t ≥ 3. In this paper, it is shown that each Z2Z4-linear Hadamard code with α = 0 is equivalent to a Z2Z4-linear Hadamard code with α 6= 0, so there are only b t 2 c nonequivalent Z2Z4-linear Hadamard codes of length 2. Moreover, the order of the monomial automorphism group for the Z2Z4-additive Hadamard codes and the permutation automorphism group of the corresponding Z2Z4-linear Hadamard codes are given.