Improved bounds and optimal constructions of pure quantum locally recoverable codes

TL;DR

This paper introduces new tighter bounds and explicit constructions for pure quantum locally recoverable codes, significantly advancing their theoretical understanding and practical design for quantum data storage.

Contribution

It provides the first set of tighter bounds for pure qLRCs and constructs many infinite families of optimal codes using classical QECCs.

Findings

New bounds for pure qLRCs are tighter than previous ones.

Explicit constructions of pure qLRCs from classical QECCs.

Identification of infinite families of optimal qLRCs with large code lengths.

Abstract

By incorporating the concept of locality into quantum information theory, quantum locally recoverable codes (qLRCs) have been proposed, motivated by their potential applications in large-scale quantum data storage and their relevance to quantum LDPC codes. Despite the progress in optimal quantum error-correcting codes (QECCs), optimal constructions of qLRCs remain largely unexplored, partly due to the fact that the existing bounds for qLRCs are not sufficiently tight. In this paper, we focus on pure qLRCs derived from the Hermitian construction. We provide several new bounds for pure qLRCs and demonstrate that they are tighter than previously known bounds. Moreover, we show that a variety of classical QECCs, including quantum Hamming codes, quantum GRM codes, and quantum Solomon-Stiffler codes, give rise to pure qLRCs with explicit parameters. Based on these constructions, we further identify many infinite families of optimal qLRCs with respect to different bounds, achieving code lengths much larger than those of known optimal qLRCs.