标题： 从射影丛中获得的局部可恢复代数几何码 显示英文标题

标题： Locally recoverable algebro-geometric codes from projective bundles

摘要： 当码字中的每个符号都可以作为$r$个其他符号的函数进行重建时，该码是局部可恢复的。 我们使用线上的射影空间束来构建具有可用性的局部可恢复码；即，每个码字符号可以从几个不相交的其他符号集合中进行重建的评估码。 最简单的情况，其中码的底层簇是一个平面，表现出值得注意的特性： 当$r = 1$,$2$,$3$时，它们是最优的；当$r \geq 4$时，随着字母表大小的增长，它们以接近$1$的概率是最优的。 此外，它们的信息率接近理论极限。 在高维情况下，我们的码形成了一族渐近好的码。 显示英文摘要

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.