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

TL;DR

This paper explores Goppa codes derived from maximal curves over finite fields, analyzing their properties, proposing improvements, and validating their performance through extensive simulations in quantum error correction contexts.

Contribution

It introduces systematic construction methods for quantum Goppa codes from maximal curves and evaluates their performance with simulations, highlighting their potential advantages and trade-offs.

Findings

Codes show promising error correction performance in simulations.

Trade-offs exist between code parameters and error correction capabilities.

Constructed codes provide systematic design approaches for quantum error correction.

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.