Quotient Space Quantum Codes

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.