Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.

翻译：利用投影Reed-Muller码构造了长量子码。投影Reed-Muller码是通过在射影空间上评估齐次多项式而得到的求值码。我们分别利用CSS构造和Hermitian构造获得了非对称与对称量子码。通过考虑等价码，我们从投影Reed-Muller码出发，构造了具有灵活纠缠量的纠缠辅助量子纠错码。此外，我们还从投影Reed-Muller码的子域子码出发，构造了量子码。