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Locally recoverable algebro-geometric codes from projective bundles

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

TL;DR

The work addresses building locally recoverable codes with availability using algebro-geometric methods. It introduces a base construction on the affine plane and a higher-dimensional generalization via projective-space bundles, achieving optimal or near-optimal distance for small locality and producing asymptotically good families as the alphabet grows or the dimension increases. Key techniques include evaluation codes, Vandermonde-based locality arguments, matroid theory to relate distance across base changes, and Lang–Weil-type counts to establish probabilistic optimality for large $q$. The results yield practical LRCs with high minimum distance and favorable rates, applicable to distributed storage, and provide a framework for extending to higher-dimensional bundles with provable asymptotic goodness. Overall, the paper blends algebraic geometry and coding theory to produce scalable, availability-rich LRCs with strong theoretical guarantees and practical relevance.

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.

Locally recoverable algebro-geometric codes from projective bundles

TL;DR

The work addresses building locally recoverable codes with availability using algebro-geometric methods. It introduces a base construction on the affine plane and a higher-dimensional generalization via projective-space bundles, achieving optimal or near-optimal distance for small locality and producing asymptotically good families as the alphabet grows or the dimension increases. Key techniques include evaluation codes, Vandermonde-based locality arguments, matroid theory to relate distance across base changes, and Lang–Weil-type counts to establish probabilistic optimality for large . The results yield practical LRCs with high minimum distance and favorable rates, applicable to distributed storage, and provide a framework for extending to higher-dimensional bundles with provable asymptotic goodness. Overall, the paper blends algebraic geometry and coding theory to produce scalable, availability-rich LRCs with strong theoretical guarantees and practical relevance.

Abstract

A code is locally recoverable when each symbol in one of its code words can be reconstructed as a function of 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 , , , they are optimal; when , they are optimal with probability approaching 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.
Paper Structure (17 sections, 23 theorems, 106 equations, 3 figures)

This paper contains 17 sections, 23 theorems, 106 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Algebro-geometric context
  3. Results
  4. Preliminaries
  5. Linear locally recoverable codes
  6. Algebro-geometric evaluation codes
  7. Codes from line bundles and certain baseline codes
  8. Set-up
  9. Relations to existing codes
  10. Dimension and locality of the code
  11. Minimum distance of the code for small locality
  12. Minimum distance of the code for localities greater than or equal to 4
  13. A detour through matroids
  14. Constructing codes with larger minimum distance
  15. Exploring nonoptimal codes
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $r = 1$, $2$, or $3$ and let $b$ satisfy $3 \leq b \leq \frac{q}{r+1}$. The code $\mathcal{C}$ is optimal and locally recoverable with locality $r$. Its parameters $[n, k, d]_q$ are

Figures (3)

  • Figure 1: The $n=b(r+1)$ points in $\mathcal{P}$.
  • Figure 2: Polynomial vanishing in the first batch.
  • Figure 3: The $b(tr+1)$ points in $\mathbb{P}^{1}\times \mathbb{P}^{m}$.

Theorems & Definitions (52)

  • Theorem 1.1: see Theorem \ref{['theo:line-fiber-optimal']}
  • Theorem 1.2: see Theorem \ref{['theo:optimal-limit']}
  • Theorem 1.3
  • Theorem 2.1: Lang--Weil estimate LangWeil54*Lemma 1
  • Example 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof
  • Corollary 3.5
  • ...and 42 more