随机代数几何码逼近插入和删除的半数Singleton界
Random Algebraic Geometry Codes Approach the Half-Singleton Bound for Insertions and Deletions
- School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究随机代数几何码在对抗性插入删除错误下的性能，通过扩展概率分析至AG码，利用不同曲线在更小域上逼近半数Singleton界，分别实现线性、次线性和独立于码长的域大小下的近乎最优纠错。
AI中文摘要：
本文研究了代数几何（AG）码在对抗性插入-删除（insdel）错误下的性能。半数Singleton界指出，一个$[n,k]_q$线性码最多能纠正$n-2k+1$个insdel错误。最近有证明表明，随机Reed-Solomon码逼近该界。然而，这些构造要求域大小$q$随码长$n$线性增长。我们通过将一般线性insdel码的概率分析扩展到AG码，克服了这一障碍。我们证明，具有许多有理点的曲线允许在显著更小的字母表上实现近乎最优的码。我们证明了以下主要渐近结果：（1）对于固定亏格的一般光滑完全曲线，随机AG码是近乎最优的，即它们能在线性大小的域（$q=\Theta(n)$）上以高概率纠正$(1-\varepsilon)n-2k$个insdel错误。（2）通过利用Hermitian曲线，我们在大小为$q=\Theta(n^{2/3})$的次线性域上实现了这种最优性，打破了线性域大小的障碍。（3）利用渐近最优的García-Stichtenoth塔，我们证明了在大小为$q=2^{O_R(1/\varepsilon^2)}$（与$n$无关）的域上，存在以高概率逼近半数Singleton界的随机AG码。
英文摘要：
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 ($q=Θ(n)$). (2) By utilizing Hermitian curves, we achieve this optimality over sublinear fields of size $q=Θ(n^{2/3})$, breaking the linear field size barrier. (3) Using asymptotically optimal García-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 $n$.
