By solving a problem regarding polynomials in a quotient ring, we obtain the relative hull and the Hermitian hull of projective Reed-Muller codes over the projective plane. The dimension of the hull determines the minimum number of maximally entangled pairs required for the corresponding entanglement-assisted quantum error-correcting code. Hence, by computing the dimension of the hull we now have all the parameters of the symmetric and asymmetric entanglement-assisted quantum error-correcting codes constructed with projective Reed-Muller codes over the projective plane. As a byproduct, we also compute the dimension of the Hermitian hull for affine Reed-Muller codes in 2 variables.

翻译：通过解决商环中的多项式问题，我们得到了射影平面上射影Reed-Muller码的相对包和Hermitian包。包的维数决定了对应纠缠辅助量子纠错码所需的最大纠缠对的最小数量。因此，通过计算包的维数，我们现在得到了利用射影平面上的射影Reed-Muller码构造的对称与非对称纠缠辅助量子纠错码的所有参数。作为副产品，我们还计算了双变量仿射Reed-Muller码的Hermitian包的维数。