Goppa Codes: Key to High Efficiency and Reliability in Communications
Behrooz Mosallaei, Farzaneh Ghanbari, Sepideh Farivar, Vahid Nourozi
TL;DR
The work addresses achieving high-efficiency, reliable communication on noisy power lines by leveraging algebraic-geometry codes from maximal curves. It constructs Goppa codes on maximal Hermitian curves $\mathcal X$ over $\mathbb F_{q^2}$, derives dimension formulas and the Hermitian self-orthogonality condition $2m\le n+2g-2$ to enable quantum stabilizer codes with parameters $[[n,n-2k,d]]_q$, and provides concrete parameter regimes for $q=3$ and $q=5$; Specifically, $n\le \#\mathcal X(\mathbb F_{q^2}) \le \tfrac{3}{2}(q^2+1)$ and $g=\tfrac{(q-1)^2}{4}$. The results show that these codes can achieve reliable high-rate transmission with minimum dual distances $d^{\perp}$, approaching Singleton-like bounds in noisy power-line channels. Practical implications include improved bandwidth for smart grids and IoT communications, with explicit constructions demonstrated for small field sizes. The work highlights a viable pathway to combining AG codes with quantum stabilizer constructions for robust, high-throughput communication.
Abstract
In this paper, we study some codes of algebraic geometry related to certain maximal curves. Quantum stabilizer codes obtained through the self orthogonality of Hermitian codes of this error correcting do not always have good parameters. However, appropriate parameters found that the Hermitian self-orthogonal code quantum stabilizer code has good parameters. Therefore, we investigated the quantum stabilizer code at a certain maximum curve and modified its parameters. Algebraic geometry codes show promise for enabling high data rate transmission over noisy power line communication channels.