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.

翻译：加性码和一些非加性码分别利用稳定子群G的单一不变子空间和多重不变子空间来构造量子码，因此不变子空间的选择是一个关键问题。本文给出了该问题的充要条件，并首次建立了用于构造量子码的商空间码。这种新型码统一了加性码和码字稳定码，且能够传输经典码字。同时给出了量子码的新界，并提供了量子Singleton界的一个简洁证明。商空间方法为量子纠错码的研究提供了一种简洁清晰的数学形式。