Algebraic Geometry Codes Approach the Half-Singleton Bound with Constant Field Size

2026-09-04

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研究通过评估码分析插入和删除错误的线性码，提出一种随机穿孔评估码的一般框架，并证明了在常数域大小下代数几何码可逼近半单界。

Abstract

We study linear codes for insertion and deletion (insdel) errors through the lens of evaluation codes. We develop a general framework for analyzing random puncturings of evaluation codes, where the edit distance is controlled by only the size of the evaluation domain and the maximum number of zeros of a nonzero function in the underlying function space. Our proof generalizes the results of Con, Guo, Li, and Zhang (ICALP 2025), and simultaneously simplifies their arguments by avoiding an in-depth analysis of longest common subsequences. We demonstrate the applicability of our core theorem by instantiating it with random puncturings of Reed--Muller codes. We then recover the result that random Reed--Solomon codes approach the half-Singleton bound over linear-sized fields while also improving the dependence on the additive gap $\varepsilon$ from $2^{O(1/\varepsilon^2)}$ to $2^{O(1/\varepsilon)}$. Finally, by applying the framework to algebraic geometry codes arising from asymptotically good towers of function fields, we show that there exist randomized families of structured linear codes over constant-sized fields that approach the half-Singleton bound.

Problem

• linear codes
• insertion and deletion errors
• evaluation codes
• half-Singleton bound
• constant field size

Innovation

• evaluation codes
• random puncturings
• half-Singleton bound
• algebraic geometry codes