Hulls of projective Reed-Muller codes over the projective plane
Information Theory
2025-03-03 v3 Commutative Algebra
math.IT
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.
Keywords
Cite
@article{arxiv.2312.13921,
title = {Hulls of projective Reed-Muller codes over the projective plane},
author = {Diego Ruano and Rodrigo San-José},
journal= {arXiv preprint arXiv:2312.13921},
year = {2025}
}