Locally recoverable algebro-geometric codes from projective bundles

TL;DR

This paper introduces a new class of locally recoverable algebraic-geometric codes constructed from projective bundles, achieving optimality and high information rates, especially in low-dimensional cases.

Contribution

It presents a novel construction of locally recoverable codes using projective bundles, demonstrating optimality and near-optimality across various parameters.

Findings

Codes are optimal for r=1, 2, 3.

Codes are asymptotically optimal as alphabet size grows for r ≥ 4.

Codes in higher dimensions are asymptotically good.

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.