Hulls of Projective Reed–Muller Codes over the Projective Plane
Diego Ruano, Rodrigo San–José · SIAM Journal on Applied Algebra and Geometry · 2024
Abstract. 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 two variables.