On Z2k-Dual Binary Codes

A new generalization of the Gray map is introduced. The new generalization Φ : Z 2 → Z k−1 n 2 is connected with the known generalized Gray map φ in the following way: if we take two dual linear Z2k -codes and construct binary codes from them using the generalizations φ and Φ of the Gray map, then the weight enumerators of the binary codes obtained will satisfy the MacWilliams identity. The classes of Z2k -linear Hadamard codes and co-Z2k-linear extended 1-perfect codes are described, where co-Z2k-linearity means that the code can be obtained from a linear Z2k -code with the help of the new generalized Gray map. Index Terms Gray map, Hadamard codes, MacWilliams identity, perfect codes, Z2k -linearity