On ℤ<sub>2</sub>ℤ<sub>4</sub>ℤ<sub>8</sub>-Additive Hadamard Codes

The ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -additive codes are subgroups of $\mathbb{Z}_2^{{\alpha _1}} \times \mathbb{Z}_4^{{\alpha _2}} \times \mathbb{Z}_8^{{\alpha _3}}$. A ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -linear Hadamard code is a Hadamard code which is the Gray map image of a ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -additive code. In this paper, we generalize some known results for ${\mathbb{Z}_2}{\mathbb{Z}_4}$-linear Hadamard codes to ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -linear Hadamard codes with ${\alpha _1} \ne 0$, ${\alpha _2} \ne 0$, and ${\alpha _3} \ne 0$. First, we give a recursive construction of ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -additive Hadamard codes of type $\left( {{\alpha _1},{\alpha _2},{\alpha _3};{t_1},{t_2},{t_3}} \right)$ with ${t_1} \geq 1,{t_2} \geq 0$, and ${t_3} \geq 1$. Then, we show for which types the corresponding ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -linear Hadamard codes are nonlinear over ${\mathbb{Z}_2}$. Moreover, we show that, unlike ${\mathbb{Z}_2}{\mathbb{Z}_4}$-linear Hadamard codes, in general, this family of ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -linear Hadamard codes does not include the family of ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> -linear or ${\mathbb{Z}_8}$-linear Hadamard codes. Actually, we show that, for example, for length ${2^{11}}$, the constructed nonlinear ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</inf> ℤ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</inf> -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 ${\mathbb{Z}_{{2^s}}}$-linear Hadamard code, with $s \geq 2$.