局部可恢复的代数几何码从投影包
Locally recoverable algebro-geometric codes from projective bundles
AI总结：
本文提出了一种基于投影包构造的局部可恢复代数几何码，展示了其在不同维度下的最优性和高信息速率特性。
摘要：
一种码是局部可恢复的，当其每个符号可以由r个其他符号重新构建。我们使用覆盖直线的投影空间包来构造具有可用性的局部可恢复码；即评估码，其中每个码字符号可以由几个不相交的其他符号集重新构建。最简单的情况，即码的底层变元是平面时，表现出值得注意的性质：当r=1、2、3时，它们是最佳的；当r≥4时，随着字母表大小的增长，它们在概率上接近最佳。此外，它们的信息速率接近理论极限。在更高维度中，我们的码形成了一族渐近良好的码。
英文摘要：
A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of $r$ other symbols. We use bundles of projective spaces over a line to construct locally recoverable codes with availability; that is, evaluation codes where each code word symbol can be reconstructed from several disjoint sets of other symbols. The simplest case, where the code's underlying variety is a plane, exhibits noteworthy properties: When $r = 1$, $2$, $3$, they are optimal; when $r \geq 4$, they are optimal with probability approaching $1$ as the alphabet size grows. Additionally, their information rate is close to the theoretical limit. In higher dimensions, our codes form a family of asymptotically good codes.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
