  Mapping an error syndrome to the error operator is the core of quantum
decoding network and is also the key step of recovery. The definitions of the
bit-flip error syndrome matrix and the phase-flip error syndrome matrix were
presented, and then the error syndromes of quantum errors were expressed in
terms of the columns of the bit-flip error syndrome matrix and the phase-flip
error syndrome matrix. It also showed that the error syndrome matrices of a
stabilizer code are determined by its check matrix, which is similar to the
classical case. So, the error-detection and recovery techniques of classical
linear codes can be applied to quantum stabilizer codes after some
modifications. Some necessary and/or sufficient conditions for the stabilizer
code over GF(2) is degenerate or nondegenerate for Pauli channel based on the
relationship between the error syndrome matrices and the check matrix was
presented. A new way to find the minimum distance of the quantum stabilizer
codes based on their check matrices was presented, and followed from which we
proved that the performance of degenerate quantum code outperform (at least
have the same performance) nondegenerate quantum code for Pauli channel.
