Quantum $(r,\delta)$-Locally Recoverable BCH and Homothetic-BCH Codes

2026-01-30

AI Summary

This work addresses the challenge of multi-node failure recovery in large-scale distributed and cloud storage systems under quantum settings by proposing a novel construction method based on BCH codes satisfying Euclidean or Hermitian dual-containing conditions and their isomorphic structures. It presents the first systematic framework for constructing pure quantum error-correcting codes with $(r,\delta)$-locality, achieving parameters that meet the Singleton-like bound with optimality. The study not only extends the theoretical pathway for translating classical locally recoverable codes into the quantum domain but also provides an efficient and scalable coding solution for highly fault-tolerant quantum storage systems.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Quantum $(r,\delta)$-locally recoverable codes ($(r,\delta)$-LRCs) are the quantum version of classical $(r,\delta)$-LRCs designed to recover multiple failures in large-scale distributed and cloud storage systems. A quantum $(r,\delta)$-LRC, $Q(C)$, can be constructed from an $(r,\delta)$-LRC, $C$, which is Euclidean or Hermitian dual-containing. This article is devoted to studying how to get quantum $(r,\delta)$-LRCs from BCH and homothetic-BCH codes. As a consequence, we give pure quantum $(r,\delta)$-LRCs which are optimal for the Singleton-like bound.

Problem

• quantum locally recoverable codes
• BCH codes
• homothetic-BCH codes
• multiple failure recovery
• distributed storage systems

Innovation

• quantum locally recoverable codes
• BCH codes
• homothetic-BCH codes
• Singleton-like bound
• dual-containing codes