On $$\mathbb {Z}_{2^s}$$ Z 2 s -linear Hadamard codes: kernel and partial classification

Abstract

The $$\mathbb {Z}_{2^s}$$ -additive codes are subgroups of $$\mathbb {Z}^n_{2^s}$$ , 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 of length $$2^t$$ 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.