${\mathbb{Z}}_2{\mathbb{Z}}_4{\mathbb{Z}}_8$-Additive Hadamard Codes

Abstract

The ${\mathbb{Z}}_2{\mathbb{Z}}_4{\mathbb{Z}}_8$-additive codes are subgroups of ${\mathbb{Z}}_2^{{\alpha }_1}\times {\mathbb{Z}}_4^{{\alpha }_2}\times {\mathbb{Z}}_8^{{\alpha }_3}$, and can be seen as linear codes over ${\mathbb{Z}}_2$ when ${\alpha }_2={\alpha }_3=0$, ${\mathbb{Z}}_4$-additive or ${\mathbb{Z}}_8$-additive codes when ${\alpha }_1={\alpha }_3=0$ or ${\alpha }_1={\alpha }_2=0$, respectively, or ${\mathbb{Z}}_2{\mathbb{Z}}_4$-additive codes when ${\alpha }_3=0$. 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 ${\alpha }_1\ne 0$, ${\alpha }_2\ne 0$, and ${\alpha }_3\ne 0$. 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 $t_1\ge 1$, $t_2\ge 0$, and $t_3\ge 1$. Then, we show that in general the ${\mathbb{Z}}_4$-linear, ${\mathbb{Z}}_8$-linear and ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear Hadamard codes are not included in the family of ${\mathbb{Z}}_2{\mathbb{Z}}_4{\mathbb{Z}}_8$-linear Hadamard codes with ${\alpha }_1\ne 0$, ${\alpha }_2\ne 0$, and ${\alpha }_3\ne 0$. Actually, we point out that none of these nonlinear ${\mathbb{Z}}_2{\mathbb{Z}}_4{\mathbb{Z}}_8$-linear Hadamard codes of length $2^{11}$ is equivalent to a ${\mathbb{Z}}_2{\mathbb{Z}}_4{\mathbb{Z}}_8$-linear Hadamard code of any other type, a ${\mathbb{Z}}_2{\mathbb{Z}}_4$-linear Hadamard code, or a ${\mathbb{Z}}_{2^s}$-linear Hadamard code, with $s\ge 2$, of the same length $2^{11}$.