Optimal Quantum $(r,\delta)$-Locally Repairable Codes via Classical Ones

TL;DR

This paper characterizes and constructs optimal quantum locally repairable codes (LRCs) with parameters $(r, \, \delta)$, establishing their relation to classical codes and providing infinite families of such codes.

Contribution

It introduces a unified decomposition theorem for optimal $(r,\delta)$-LRCs, characterizes their quantum counterparts, and constructs infinite families of optimal quantum LRCs.

Findings

Local protection codes are MDS codes with the same minimum Hamming distance.

Minimum Hamming distance of optimal $(r,\delta)$-LRCs satisfies $d \geq \delta$.

Constructed three infinite families of optimal quantum $(r,\delta)$-LRCs.

Abstract

Locally repairable codes (LRCs) play a crucial role in mitigating data loss in large-scale distributed and cloud storage systems. This paper establishes a unified decomposition theorem for general optimal $(r,\delta)$-LRCs. Based on this, we obtain that the local protection codes of general optimal $(r,\delta)$-LRCs are MDS codes with the same minimum Hamming distance $\delta$. We prove that for general optimal $(r,\delta)$-LRCs, their minimum Hamming distance $d$ always satisfies $d\geq \delta$. We fully characterize the optimal quantum $(r,\delta)$-LRCs induced by classical optimal $(r,\delta)$-LRCs that admit a minimal decomposition. We construct three infinite families of optimal quantum $(r,\delta)$-LRCs with flexible parameters.