Long quantum codes using projective Reed-Muller codes are constructed. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. Quantum codes obtained from projective Reed-Muller codes usually require entanglement assistance, but we show that sometimes we can avoid this requirement by considering monomially equivalent codes. Moreover, we also provide some constructions of quantum codes from subfield subcodes of projective Reed-Muller codes.

翻译：利用射影Reed-Muller码构建了长量子码。通过CSS构造和Hermitian构造，分别获得了非对称与对称量子码。从射影Reed-Muller码得到的量子码通常需要纠缠辅助，但本文证明，通过考虑单项式等价码，有时可以避免这一需求。此外，我们还提供了基于射影Reed-Muller码的子域子码构建量子码的若干方法。