Progress on Problem about Quantum Hamming Bound for Impure Quantum Codes

A famous open problem in the theory of quantum error-correcting codes is whether or not the parameters of an impure quantum code can violate the quantum Hamming bound for pure quantum codes. We partially solve this problem. We demonstrate that there exists a threshold such that an arbitrary quantum code must obey the quantum Hamming bound whenever . We list some values of for small d and binary quantum codes.