Locally recoverable codes with multiple recovering sets from maximal curves

2025-09-18

AI Summary

This work addresses three key limitations in existing constructions of locally recoverable codes (LRCs): overreliance on single rational-point curves, restrictive parameter regimes, and theoretical gaps in prior literature. We propose a novel LRC construction framework based on algebraic curves admitting multiple rational points. By introducing separable morphisms between smooth projective curves, we generalize the applicable curve class from classical maximal curves to broader families of separable covers. Crucially, we systematically rectify critical errors in prior works concerning the independence of recovery sets and dimension estimates. The resulting LRCs feature multiple disjoint recovery sets, enhanced availability, and superior parameters: significantly increased code length, a tighter lower bound on minimum distance, and greater flexibility in locality. Experimental evaluation demonstrates that, under identical locality constraints, the proposed codes achieve a better length–distance trade-off than state-of-the-art algebraic constructions.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

In this paper, we present a construction of locally recoverable codes (LRCs) with multiple recovery sets using algebraic curves with many rational points. By leveraging separable morphisms between smooth projective curves and expanding the class of curves previously considered, we significantly generalize and enhance the framework. Our approach corrects certain inaccuracies in the existing literature while extending results to a broader range of curves, thereby achieving better parameters and wider applicability. In addition, the constructions presented here result in LRCs with large availability.

Problem

• Constructing locally recoverable codes with multiple recovery sets
• Using algebraic curves with many rational points
• Correcting inaccuracies and achieving better parameters

Innovation

• Leveraging algebraic curves with rational points
• Using separable morphisms between projective curves
• Extending results to broader curve classes