  In quantum coding theory, stabilizer codes are probably the most important
class of quantum codes. They are regarded as the quantum analogue of the
classical linear codes and the properties of stabilizer codes have been
carefully studied in the literature. In this paper, a new but simple
construction of stabilizer codes is proposed based on syndrome assignment by
classical parity-check matrices. This method reduces the construction of
quantum stabilizer codes to the construction of classical parity-check matrices
that satisfy a specific commutative condition. The quantum stabilizer codes
from this construction have a larger set of correctable error operators than
expected. Its (asymptotic) coding efficiency is comparable to that of CSS
codes. A class of quantum Reed-Muller codes is constructed, which have a larger
set of correctable error operators than that of the quantum Reed-Muller codes
developed previously in the literature. Quantum stabilizer codes inspired by
classical quadratic residue codes are also constructed and some of which are
optimal in terms of their coding parameters.
