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.

翻译：若码的每个码字符号均可由$r$个其他符号的函数重构，则称该码具有局部可恢复性。本文利用直线上射影空间的丛构造具有可用性的局部可恢复码——即每个码字符号可从若干互不相交的其他符号集合中重构的求值码。当码的底层簇为平面这一最简单情形时，展现出显著特性：当$r = 1$、$2$、$3$时，这些码是最优的；当$r \geq 4$时，随着字母表规模增大，它们以趋近于$1$的概率达到最优。此外，其信息率接近理论极限。在高维情形下，我们构造的码构成一族渐进好码。