New Constructions of Optimal $(r,δ)$-LRCs via Algebraic Function Fields

2025-08-29

AI Summary

Constructing optimal (r,δ)-locally recoverable codes (LRCs) with flexible minimum distance δ ≥ 3 for distributed storage remains challenging. Method: This paper pioneers the extension of automorphism groups of elliptic function fields to (r,δ)-LRC design, introducing a novel algebraic construction framework based on high-genus hyperelliptic and supertrace curves. Leveraging automorphism groups of hyperelliptic function fields of genus g = 1, 2, and arbitrary g—combined with finite field theory and algebraic curve theory—we systematically derive explicit constructions. Results: We obtain several families of optimal LRCs, including (r,3)-, (2,δ)-, and (g+1−g′,g+1+g′)-LRCs. The constructed codes achieve lengths ranging from q+2√q to q+4√q; some attain the current longest known length, significantly broadening both parameter flexibility and achievable code length bounds.

Technology Category

• Search and Optimization: Distributed Search
• Intelligent Robots: Learning & Optimization for ROB
• Machine Learning: Deep Generative Models & Autoencoders

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Constructing optimal $(r,δ)$-LRCs that attain the Singleton-type bound is an active and important research direction, particularly due to their practical applications in distributed storage systems. In this paper, we focus on the construction of optimal $(r,δ)$-LRCs with flexible minimum distances, especially for the case $δgeq 3$. We first extend a general framework -- originally proposed by Li extit{et al.} (IEEE Trans. Inf. Theory, vol. 65, no. 1, 2019) and Ma and Xing (J. Comb. Theory Ser. A., vol. 193, 2023) -- for constructing optimal $r$-LRCs via automorphism groups of elliptic function fields to the case of $(r,δ)$-LRCs. This newly extended general framework relies on certain conditions concerning the group law of elliptic curves. By carefully selecting elliptic function fields suitable for this framework, we arrive at several families of explicit $q$-ary optimal $(r,3)$-LRCs and $(2,δ)$-LRCs with lengths slightly less than $q + 2sqrt{q}$. Next, by employing automorphism groups of hyperelliptic function fields of genus $2$, we develop a framework for constructing optimal $(r,3)$-LRCs and obtain a family of explicit $q$-ary optimal $(4,3)$-LRCs with code lengths slightly below $q+4sqrt{q}$. We then consider the construction of optimal $(r,δ)$-LRCs via hyperelliptic function fields of arbitrary genus $g geq 2$, yielding a class of explicit $q$-ary optimal $(g+1-g',g+1+g')$-LRCs for $0 leq g' leq g-1$ with lengths up to $q + 2gsqrt{q}$. Finally, applying certain superelliptic curves derived from modified Norm-Trace curves, we construct two families of explicit optimal $(r,δ)$-LRCs with even longer code lengths and more flexible parameters. Notably, many of the newly constructed optimal $(r,δ)$-LRCs attain the largest known lengths among existing constructions with flexible minimum distances.

Problem

• Constructing optimal locally recoverable codes with flexible parameters
• Extending algebraic frameworks for LRCs using elliptic curves
• Developing new LRC families via hyperelliptic and superelliptic function fields

Innovation

• Extended framework using elliptic function fields
• Employed hyperelliptic function fields genus 2
• Applied superelliptic curves from modified Norm-Trace