New entanglement-assistant and asymmetric quantum codes from quasi-cyclic codes of index two

In this paper, we systematically investigate the construction of two important classes of quantum error-correcting codes, namely entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes, from a generalized class of index-2 quasi-cyclic codes. For entanglement-assisted quantum error-correcting codes, we establish a systematic construction framework and derive sufficient conditions for obtaining maximal-entanglement entanglement-assisted quantum error-correcting codes, from which two new classes of such codes are obtained. For asymmetric quantum error-correcting codes, we derive necessary and sufficient conditions for the inclusion relation between any two index-2 QC codes and use them to establish a systematic construction framework. Several new entanglement-assisted quantum error-correcting codes and asymmetric quantum error-correcting codes with parameters improving upon previously reported results are obtained as applications of the proposed constructions.