  Additive codes and some nonadditive codes use the single and multiple
invariant subspaces of the stabilizer G to construct quantum codes,
respectively, so the selection of invariant subspaces is a key issue. In this
letter, I provide the necessary and sufficient conditions for this problem and,
for the first time, establish the quotient space codes to construct quantum
codes. This new code unifies additive codes and codeword stabilized codes and
can transmit classical codewords. New bounds for quantum codes are presented
also, and a simple proof of the quantum Singleton bound is provided. The
quotient space approach offers a concise and clear mathematical form for the
study of quantum error-correcting codes.
