Hulls of projective Reed-Muller codes over the projective plane

TL;DR

This work determines the relative Euclidean and Hermitian hulls of projective Reed-Muller codes on the projective plane ${\mathbb P}^2$, and shows how hull dimensions directly fix the entanglement parameter $c$ in entanglement-assisted QECCs. By translating evaluation codes into the quotient ring $S/I({\mathbb P}^2)$ and applying Gröbner-bases techniques, the authors construct explicit bases for hulls and provide comprehensive dimension formulas across parameter regimes, including detailed Euclidean and Hermitian cases and their dualities. The results yield concrete EAQECC parameters with long lengths and favorable distances, including numerous examples that surpass quantum GV bounds and exhibit high asymmetry in error correction. As a byproduct, they also obtain the Hermitian hull dimensions for affine Reed-Muller codes in two variables, and they present exact or sharp bounds for the code parameters, enabling direct construction of pure, entanglement-assisted quantum codes from PRM codes on ${\mathbb P}^2$.

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.

Theorems & Definitions (76)

• Theorem 2.1
• Theorem 2.2
• Remark 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Lemma 2.7
• proof
• Remark 2.8
• Lemma 2.9