发表机构
Aalto University（阿尔托大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究开发评价码随机删截分析框架，推广简化相关成果，将加性间隙ε的依赖改进为2^{O(1/ε)}，证明有限域大小下存在逼近半单利边界的结构化线性码随机族。
AI 中文摘要
我们通过评价码的视角研究可纠正插入和删除（insdel）错误的线性码。我们开发了一个通用框架，用于分析评价码的随机删截，其中编辑距离仅由评价域的大小和底层函数空间中非零函数的最大零点数控制。我们的证明推广了Con、Guo、Li和Zhang（ICALP 2025）的结果，同时通过避免对最长公共子序列的深入分析简化了他们的论证。我们通过将核心定理实例化到里德-穆勒（Reed-Muller）码的随机删截来证明其适用性。随后，我们得到以下结果：随机里德-所罗门（Reed-Solomon）码在线性大小的域上逼近半单利边界，同时将加性间隙ε的依赖关系从2^{O(1/ε²)}改进为2^{O(1/ε)}。最后，通过将该框架应用于由函数域渐近良塔产生的代数几何码，我们证明存在有限域大小下的结构化线性码随机族，可逼近半单利边界。
英文摘要
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.
