Complete Weight Distributions and MacWilliams Identities for Asymmetric Quantum Codes

In 1997, Shor and Laflamme defined the weight enumerators for quantum error-correcting codes and derived a MacWilliams identity. We extend their work by introducing our double weight enumerators and complete weight enumerators for qubit codes and then investigate the MacWilliams identities for these enumerators. Based on the generalized MacWilliams identities, we solve an open problem, namely, the Singleton-type bound for asymmetric quantum codes (AQCs). Besides, the Hamming-type and the first linear-programming-type bounds for the AQCs are deduced similarly.