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arXiv · 2409.04201

Locally recoverable algebro-geometric codes from projective bundles

Konrad Aguilar·Angelynn Álvarez·René Ardila·Pablo S. Ocal·Cristian Rodriguez Avila·Anthony Várilly-Alvarado
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Abstract

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.

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Konrad Aguilar, Angelynn Álvarez, René Ardila, Pablo S. Ocal, Cristian Rodriguez Avila, Anthony Várilly-Alvarado. 2026-01-07. Locally recoverable algebro-geometric codes from projective bundles. https://arxiv.org/abs/2409.04201

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