Classification of the -Linear Hadamard Codes and Their Automorphism Groups

A Z₂Z₄-linear Hadamard code of length α + 2β = 2^t 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 ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z₂Z₄-linear Hadamard codes of length 2^t, with α = 0 and α ≠ 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 ≠ 0, so there are only ⌊t/2⌋ nonequivalent Z₂Z₄-linear Hadamard codes of length 2^t. 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.