Random Algebraic Geometry Codes Approach the Half-Singleton Bound for Insertions and Deletions

2026-09-06

AI Summary

本文研究了代数几何码在对抗插入-删除错误时的表现，通过使用具有许多有理点的曲线，实现了在较小字母表上接近最优的性能，突破了线性字段大小限制。

Abstract

In this paper, we study the performance of algebraic geometry (AG) codes against adversarial insertion-deletion (insdel) errors. The half-Singleton bound states that an $[n,k]_q$ linear code can correct at most $n-2k+1$ insdel errors. It was recently proven that random Reed-Solomon codes approach this bound. However, these constructions require the field size $q$ to grow linearly with the code length $n$. We overcome this barrier by extending the probabilistic analysis of general linear insdel codes to AG codes. We demonstrate that curves with many rational points allow for nearly optimal codes over significantly smaller alphabets. We prove the following main asymptotic results: (1) For general smooth complete curves of fixed genus, random AG codes are nearly optimal, that is, they can correct $(1-\varepsilon)n-2k$ insdel errors with high probability over linear-sized fields ($q=\Theta(n)$). (2) By utilizing Hermitian curves, we achieve this optimality over sublinear fields of size $q=\Theta(n^{2/3})$, breaking the linear field size barrier. (3) Using asymptotically optimal Garc\'{i}a-Stichtenoth towers, we prove the existence of random AG codes that approach the half-Singleton bound with high probability over fields of size $q=2^{O_R(1/\varepsilon^2)}$, independent of $n$.

Problem

• algebraic geometry codes
• insertion-deletion errors
• half-Singleton bound
• small alphabets

Innovation

• Algebraic Geometry Codes
• Insertion-Deletion Errors
• Half-Singleton Bound
• Rational Points
• García-Stichtenoth Towers