Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On $Z_{2^k}$-Dual Binary Codes   A new generalization of the Gray map is introduced. The new generalization
$\Phi: Z_{2^k}^n \to Z_{2}^{2^{k-1}n}$ is connected with the known generalized
Gray map $\phi$ in the following way: if we take two dual linear
$Z_{2^k}$-codes and construct binary codes from them using the generalizations
$\phi$ and $\Phi$ of the Gray map, then the weight enumerators of the binary
codes obtained will satisfy the MacWilliams identity. The classes of
$Z_{2^k}$-linear Hadamard codes and co-$Z_{2^k}$-linear extended 1-perfect
codes are described, where co-$Z_{2^k}$-linearity means that the code can be
obtained from a linear $Z_{2^k}$-code with the help of the new generalized Gray
map. Keywords: Gray map, Hadamard codes, MacWilliams identity, perfect codes,
$Z_{2^k}$-linearity

16
The categories of this paper are: stat
0
The categories of this paper are: hep-ph
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
35
The categories of this paper are: mtrl-th
34
The categories of this paper are: chem-ph
27
The categories of this paper are: comp-gas
14
The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
22
The categories of this paper are: chao-dyn
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con