Locally recoverable algebro-geometric codes with multiple recovery sets from projective bundles
<|reference_start|>Locally recoverable algebro-geometric codes with multiple recovery sets from projective bundles: 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.<|reference_end|>
@article{aguilar2024locally,
    title={Locally recoverable algebro-geometric codes with multiple recovery sets
  from projective bundles},
    author={Konrad Aguilar, Angelynn 'Alvarez, Ren'e Ardila, Pablo S. Ocal,
  Cristian Rodriguez Avila, Anthony V'arilly-Alvarado},
    journal={arXiv preprint arXiv:2409.04201},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.04201},
    primaryClass={cs.IT math.AG math.IT}
}