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

A new generalization of the Gray map is introduced. The new generalization Phi:Z
₂

^kn
rarr Z
₂

^2k-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