On the complete weight distributions of quantum error-correcting codes

Abstract

Abstract In a recent paper Hu et al. defined the complete weight distributions of quantum codes and proved the MacWilliams identities, and as applications they showed how such weight distributions may be used to obtain the Singleton-type and Hamming-type bounds for asymmetric quantum codes. In this paper we extend their study much further and obtain several new results concerning the complete weight distributions of quantum codes and applications. In particular, we provide a new proof of the MacWilliams identities of the complete weight distributions of quantum codes; we obtain new information on the weight distributions of quantum MDS codes and the double weight distribution of asymmetric quantum MDS codes; we obtain new identities involving the complete weight distributions of two different quantum codes; we estimate the complete weight distributions of quantum codes under special conditions and show that quantum BCH codes by the Hermitian construction from primitive, narrow-sense BCH codes satisfy these conditions and hence these estimate applies.