Constructions of Quantum $(r,δ)$-LRCs from cyclic codes

2026-06-08

AI Summary

This work addresses the construction of quantum $(r,\delta)$-locally recoverable codes (qLRCs) from classical cyclic LRCs that satisfy the dual-containing condition, aiming to enable highly efficient fault-tolerant quantum storage. By analyzing the defining sets of cyclic codes and rigorously verifying their dual-containing property within the CSS framework, the authors systematically propose two general constructions of $(r,\delta)$-qLRCs without restrictions on code length. Among the three families of quantum codes obtained, two achieve the quantum Singleton-like bound in the pure case, establishing them as currently best-known constructions. These families are applicable to arbitrary code lengths and finite field sizes, significantly broadening the design space for high-performance quantum LRCs.

Abstract

Classical $(r,δ)$ locally recoverable codes (LRCs) play a central role in distributed data storage systems as they enable an efficient recovery from erasures by accessing a small number of surviving symbols. Motivated by their prospective use in future quantum data storage and by recent theoretical progress on quantum locally recoverable codes (qLRCs), we investigate the construction of qLRCs from classical cyclic $(r,δ)$-LRCs. Our approach identifies cyclic LRCs whose defining sets satisfy a dual-containing condition, allowing them to serve as valid CSS ingredients. We present three explicit families of $(r,δ)$-qLRCs, two of which are optimal with respect to the quantum Singleton-like bound, whenever the codes are pure, thereby providing optimal examples. Additionally, the codes presented in Constructions 2 and 3 have no bound on their lengths with respect to the field size required to obtain these codes.

Problem

• quantum locally recoverable codes
• cyclic codes
• quantum data storage
• local recovery
• CSS construction

Innovation

• quantum LRCs
• cyclic codes
• CSS construction
• optimal codes
• unbounded length