Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Kernel of $\mathbb{Z}_{2^s}$-Linear Hadamard Codes   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 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 $s\geq 2$, by using
also the rank and, in some cases, further computations.

13
The categories of this paper are: quant-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
11
The categories of this paper are: hep-lat
15
The categories of this paper are: nucl-ex
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The categories of this paper are: eess
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
4
The categories of this paper are: cond-mat
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The categories of this paper are: q-fin
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-th
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The categories of this paper are: physics
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The categories of this paper are: bayes-an
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The categories of this paper are: nucl-th