The hull variation problem for projective Reed-Muller codes and quantum error-correcting codes

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.