发表机构
School of Computer Science, Shanghai Jiao Tong University（上海交通大学计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对局部逐坐标线性（LCL）性质，研究随机单点AG码的阈值，证明其在明确条件下与随机线性码的LCL阈值率一致，消除了Reed-Solomon论证的辅助下界，还修正了随机Reed-Solomon码阈值证明的缺陷，拓展了LCL阈值框架至随机单点AG码。
AI 中文摘要
局部逐坐标线性（LCL）性质为研究线性码的列表译码、列表恢复及其他局部性质提供了统一框架。我们证明，在明确的字母表和采样条件下，随机单点代数几何（AG）码具有与随机线性码相同的LCL阈值率。更准确地说，对于阈值为$R_{\mathcal P}$的局部轮廓族，随机单点AG码会避开$R_{\mathcal P}-\varepsilon$以下的所有此类轮廓，并包含$R_{\mathcal P}+\varepsilon$以上的轮廓，两侧均有明确的失效界。我们的证明还消除了直接适配里德-所罗门（Reed–Solomon）论证时出现的关于可用有理点数的辅助下界，且无需额外的依赖亏格的率间隙。我们还发现，之前声称的随机里德-所罗门码的阈值以上证明并未建立所需的概率下界，该论证中使用的包含关系给出的概率比较方向与所需方向相反。我们给出了一种不同的阈值以上论证，该论证尤其在亏格为零的情况下也能得到另一种证明。将我们的阈值定理与随机线性码的最新结果相结合，可为随机单点AG码提供一系列新的列表译码、平均权重列表译码和列表恢复保证。我们还获得了多项式曲线的新相关一致性和邻近间隙结果。这些结果将LCL阈值框架从随机线性码和里德-所罗门码扩展到了随机单点AG码。
英文摘要
Local coordinate-wise linear (LCL) properties provide a unified framework for studying list decoding, list recovery and other local properties of linear codes. We prove that random one-point algebraic geometry (AG) codes have the same LCL threshold rates as random linear codes under explicit alphabet and sampling conditions. More precisely, for a family of local profiles with random linear code threshold $R_{\mathcal P}$, a random one-point AG code avoids all such profiles below $R_{\mathcal P}-\varepsilon$ and contains one above $R_{\mathcal P}+\varepsilon$, with explicit failure bounds on both sides. Our proof also removes an auxiliary lower bound on the number of available rational places that arises in a direct adaptation of the Reed--Solomon argument and requires no additional genus-dependent rate gap. We also observe that the previously claimed above threshold proof for random Reed--Solomon codes does not establish the required probability lower bound. The containment relation used in that argument gives the probability comparison in the opposite direction from what is needed. We give a different above threshold argument, which in particular also yields an alternative proof in the genus zero case. Combining our threshold theorem with recent results for random linear codes gives a collection of new list decoding, average weight list decoding and list recovery guarantees for random one-point AG codes. We further obtain new correlated agreement and proximity gap results for polynomial curves. These results extend the LCL threshold framework from random linear and Reed--Solomon codes to random one-point AG codes.
Comments37 pages
