Locally recoverable algebro-geometric codes from projective bundles
Konrad Aguilar, Álvarez, Angelynn, René Ardila, Pablo S. Ocal, Cristian Rodriguez Avila, Anthony Várilly‐Alvarado · arXiv (Cornell University) · 2024
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.