The application of weight parity error correction in quantum codes

Weight parity error correction (WPEC) is an error-correction scheme that can correct high-weight Pauli errors. Still, it was initially shown to apply only to Steane and Golay codes. In this work, we prove that the WPEC scheme is appropriate for concatenated codes constructed from two classes of quantum codes, quantum quadratic-residue codes and quantum stabilizer codes from self-orthogonal cyclic codes. We generalize the implementation of the WPEC technique to nonbinary quantum codes and provide an application using WPEC for the [[121, 1, 25]](3) concatenated code. We also show that the WPEC may not be applicable for some quantum codes with specific parameters through an example of the [[15, 7]] code over F-2.