Classification of the $Z_{2}Z_{4}$ -Linear Hadamard Codes and Their Automorphism Groups

A Z
₂
Z
₄
-linear Hadamard code of length α + 2β = 2
^t
is a binary Hadamard code, which is the Gray map image of a Z
₂
Z
₄
-additive code with α binary coordinates and β quaternary coordinates. It is known that there are exactly ⌊t-1/2⌋ and ⌊t/2⌋ nonequivalent Z
₂
Z
₄
-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 Z
₂
Z4-linear Hadamard code with α = 0 is equivalent to a Z
₂
Z
₄
-linear Hadamard code with α ≠ 0, so there are only ⌊t/2⌋ nonequivalent Z
₂
Z
₄
-linear Hadamard codes of length 2
^t
. Moreover, the order of the monomial automorphism group for the Z
₂
Z
₄
-additive Hadamard codes and the permutation automorphism group of the corresponding Z
₂
Z
₄
-linear Hadamard codes are given.