Hulls of projective Reed-Muller codes over the projective plane

TL;DR

This paper computes the hull dimensions of projective Reed-Muller codes over the projective plane, enabling the construction of entanglement-assisted quantum error-correcting codes with known parameters.

Contribution

It introduces a method to determine the hulls of these codes by solving polynomial problems in quotient rings, linking classical code properties to quantum code parameters.

Findings

Computed the relative and Hermitian hulls of projective Reed-Muller codes.

Determined the parameters for entanglement-assisted quantum codes from these hulls.

Also calculated the Hermitian hull dimension for affine Reed-Muller codes in two variables.

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 2 variables.