On $Z_{2^k}$-Dual Binary Codes

A new generalization of the Gray map is introduced. The new generalization $\mathrm{\Phi }:Z_{2^k}^n\to Z_2^{2^{k-1}n}$ is connected with the known generalized Gray map $\varphi$ in the following way: if we take two dual linear $Z_{2^k}$-codes and construct binary codes from them using the generalizations $\varphi$ and $\mathrm{\Phi }$ 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