On the Kernel of ${\mathbb{Z}}_{2^s}$-Linear Hadamard Codes

Abstract

The ${\mathbb{Z}}_{2^s}$-additive codes are subgroups of ${\mathbb{Z}}_{2^s}^n$, and can be seen as a generalization of linear codes over ${\mathbb{Z}}_2$ and ${\mathbb{Z}}_4$. A ${\mathbb{Z}}_{2^s}$-linear Hadamard code is a binary Hadamard code which is the Gray map image of a ${\mathbb{Z}}_{2^s}$-additive code. It is known that the dimension of the kernel can be used to give a complete classification of the ${\mathbb{Z}}_4$-linear Hadamard codes. In this paper, the kernel of ${\mathbb{Z}}_{2^s}$-linear Hadamard codes and its dimension are established for $s>2$. Moreover, we prove that this invariant only provides a complete classification for some values of $t$ and $s$. The exact amount of nonequivalent such codes are given up to $t=11$ for any $s\ge 2$, by using also the rank and, in some cases, further computations.