Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem

Diego Ruano, Rodrigo San-José

TL;DR

The paper develops long quantum error-correcting codes from projective Reed-Muller codes using both CSS (asymmetric) and Hermitian (symmetric) constructions, and introduces hull variation as a tool to tune the entanglement parameter $c$ via equivalent codes. By exploiting hulls Hull_{C_2}(C_1) and Hull_H(C) and subfield subcodes, it shows how to realize unassisted QECCs ($c=0$) and a flexible range of entanglement while maintaining the net rate. The results include explicit parameter regimes where PRM-based codes surpass the quantum Gilbert–Varshamov bound and outperform affine Reed-Muller codes, with numerous examples over small fields and long lengths. The work provides a versatile framework for designing QECCs with tailored entanglement resources, distance, and rate, applicable to practical quantum communication settings.

Abstract

Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.

Quantum error-correcting codes from projective Reed-Muller codes and their hull variation problem

TL;DR

The paper develops long quantum error-correcting codes from projective Reed-Muller codes using both CSS (asymmetric) and Hermitian (symmetric) constructions, and introduces hull variation as a tool to tune the entanglement parameter via equivalent codes. By exploiting hulls Hull_{C_2}(C_1) and Hull_H(C) and subfield subcodes, it shows how to realize unassisted QECCs () and a flexible range of entanglement while maintaining the net rate. The results include explicit parameter regimes where PRM-based codes surpass the quantum Gilbert–Varshamov bound and outperform affine Reed-Muller codes, with numerous examples over small fields and long lengths. The work provides a versatile framework for designing QECCs with tailored entanglement resources, distance, and rate, applicable to practical quantum communication settings.

Abstract

Long quantum codes using projective Reed-Muller codes are constructed. Projective Reed-Muller codes are evaluation codes obtained by evaluating homogeneous polynomials at the projective space. We obtain asymmetric and symmetric quantum codes by using the CSS construction and the Hermitian construction, respectively. We provide entanglement-assisted quantum error-correcting codes from projective Reed-Muller codes with flexible amounts of entanglement by considering equivalent codes. Moreover, we also construct quantum codes from subfield subcodes of projective Reed-Muller codes.
Paper Structure (4 sections, 19 theorems, 32 equations)

This paper contains 4 sections, 19 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. CSS construction
  4. Hermitian construction

Key Result

Theorem 2.1

The projective Reed-Muller code $\mathop{\mathrm{PRM}}\nolimits_d(q,m)$, $1\leq d\leq m(q-1)$, is an $[n,k]$-code with For the minimum distance, we have

Theorems & Definitions (37)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5: (CSS Construction)
  • Remark 2.6
  • Theorem 2.7: (Hermitian construction)
  • Theorem 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 27 more