On recursive constructions of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes

The $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive codes are subgroups of $ \mathbb{Z}_2^{\alpha_1} \times \mathbb{Z}_4^{\alpha_2} \times \mathbb{Z}_8^{\alpha_3} $. A $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard code is a Hadamard code, which is the Gray map image of a $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive code. In this paper, we generalize some known results for $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard codes to $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes with $ \alpha_1 \neq 0 $, $ \alpha_2 \neq 0 $, and $ \alpha_3 \neq 0 $. First, we give a recursive construction of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive Hadamard codes of type $ (\alpha_1, \alpha_2, \alpha_3;t_1, t_2, t_3) $ with $ t_1\geq 1 $, $ t_2 \geq 0 $, and $ t_3\geq 1 $. It is known that each $ \mathbb{Z}_4 $-linear Hadamard code is equivalent to a $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard code with $ \alpha_1\neq 0 $ and $ \alpha_2\neq 0 $. Unlike $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard codes, in general, this family of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes does not include the family of $ \mathbb{Z}_4 $-linear or $ \mathbb{Z}_8 $-linear Hadamard codes. We show that, for example, for length $ 2^{11} $, the constructed nonlinear $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-linear Hadamard codes are not equivalent to each other, nor to any $ \mathbb{Z}_2 \mathbb{Z}_4 $-linear Hadamard, nor to any previously constructed $ \mathbb{Z}_{2^s} $-Hadamard code, with $ s\geq 2 $. Finally, we also present other recursive constructions of $ \mathbb{Z}_2 \mathbb{Z}_4 \mathbb{Z}_8 $-additive Hadamard codes having the same type, and we show that, after applying the Gray map, the codes obtained are equivalent to the previous ones.