On $Z_2^k$-Dual Binary Codes
<|reference_start|>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<|reference_end|>
@article{krotov2005on,
    title={On $Z_{2^k}$-Dual Binary Codes},
    author={Denis Krotov (Sobolev Institute of Mathematics, Novosibirsk, Russia)},
    journal={IEEE Trans. Inf. Theory 53(4) 2007, 1532-1537},
    year={2005},
    doi={10.1109/TIT.2007.892787},
    archivePrefix={arXiv},
    eprint={math/0509325},
    primaryClass={math.CO cs.IT math.IT}
}