Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Error Correction with Goppa Codes from Maximal Curves: Design, Simulation, and Performance

Vahid Nourozi

TL;DR

This work addresses the construction and analysis of Goppa codes from a family of maximal curves defined by $y^n = x^m + x$ and exploits Hermitian self-orthogonality to obtain quantum stabilizer codes. It develops explicit bases for Riemann-Roch spaces, establishes duality and self-orthogonality criteria, and provides concrete code parameters for both classical and quantum regimes, supplemented by simulation results. The paper demonstrates that, although the resultant minimum distances are not always optimal, the approach yields systematic, parameter-rich families of codes with favorable trade-offs between length, dimension, and error-correcting capability. The contributions advance quantum code construction from a concrete algebraic-geometry framework and offer guidance for future optimization and application-specific code design.

Abstract

This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form $y^n = x^m + x$. We investigate Algebraic Geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.

Quantum Error Correction with Goppa Codes from Maximal Curves: Design, Simulation, and Performance

TL;DR

This work addresses the construction and analysis of Goppa codes from a family of maximal curves defined by and exploits Hermitian self-orthogonality to obtain quantum stabilizer codes. It develops explicit bases for Riemann-Roch spaces, establishes duality and self-orthogonality criteria, and provides concrete code parameters for both classical and quantum regimes, supplemented by simulation results. The paper demonstrates that, although the resultant minimum distances are not always optimal, the approach yields systematic, parameter-rich families of codes with favorable trade-offs between length, dimension, and error-correcting capability. The contributions advance quantum code construction from a concrete algebraic-geometry framework and offer guidance for future optimization and application-specific code design.

Abstract

This paper characterizes Goppa codes of certain maximal curves over finite fields defined by equations of the form . We investigate Algebraic Geometric and quantum stabilizer codes associated with these maximal curves and propose modifications to improve their parameters. The theoretical analysis is complemented by extensive simulation results, which validate the performance of these codes under various error rates. We provide concrete examples of the constructed codes, comparing them with known results to highlight their strengths and trade-offs. The simulation data, presented through detailed graphs and tables, offers insights into the practical behavior of these codes in noisy environments. Our findings demonstrate that while the constructed codes may not always achieve optimal minimum distances, they offer systematic construction methods and interesting parameter trade-offs that could be valuable in specific applications or for further theoretical study.
Paper Structure (9 sections, 8 theorems, 26 equations, 2 figures, 3 algorithms)

This paper contains 9 sections, 8 theorems, 26 equations, 2 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Algebraic Geometry Codes
  3. Goppa Code Over Curve $X$
  4. Simulation Results
  5. Simulation Methodology
  6. Algorithms
  7. Results and Analysis
  8. Quantum Stabilizer Codes Over Curve $X$
  9. Conclusion

Key Result

Lemma 3.3

Let $X$ be the curve defined as above, and let $D$ and $G$ be divisors as described. Then: where $K = div(\eta) \in Div_q(X)$ is a canonical divisor defined by a differential $\eta$ such that $\nu_{P_i}(\eta) = -1$ and $res_{P_i}(\eta) = 1$ for each $i = 1, 2, \ldots, n$.

Figures (2)

  • Figure 1: Goppa Code Performance (Individual Codes)
  • Figure 2: Average Errors vs Error Rate

Theorems & Definitions (21)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 3.1
  • Example 3.2
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • ...and 11 more