Locally recoverable algebro-geometric codes from projective bundles
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arXiv
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| Auteurs principaux: | , , , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912808515403776 |
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| author | Aguilar, Konrad Álvarez, Angelynn Ardila, René Ocal, Pablo S. Avila, Cristian Rodriguez Várilly-Alvarado, Anthony |
| author_facet | Aguilar, Konrad Álvarez, Angelynn Ardila, René Ocal, Pablo S. Avila, Cristian Rodriguez Várilly-Alvarado, Anthony |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Locally recoverable algebro-geometric codes from projective bundles Aguilar, Konrad Álvarez, Angelynn Ardila, René Ocal, Pablo S. Avila, Cristian Rodriguez Várilly-Alvarado, Anthony Information Theory Algebraic Geometry 94B27, 14G50, 11G25 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. |
| title | Locally recoverable algebro-geometric codes from projective bundles |
| topic | Information Theory Algebraic Geometry 94B27, 14G50, 11G25 |
| url | https://arxiv.org/abs/2409.04201 |