Optimal Quantum $(r,\hat{I}´)$-Locally Repairable Codes via Classical Ones

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 $(r,\hat{I}´)$-LRCs. Based on this, we obtain that the local protection codes of general optimal $(r,\hat{I}´)$-LRCs are MDS codes with the same minimum Hamming distance $\hat{I}´$. We prove that for general optimal $(r,\hat{I}´)$-LRCs, their minimum Hamming distance $d$ always satisfies $d\ge \hat{I}´$. We fully characterize the optimal quantum $(r,\hat{I}´)$-LRCs induced by classical optimal $(r,\hat{I}´)$-LRCs that admit a minimal decomposition. We construct three infinite families of optimal quantum $(r,\hat{I}´)$-LRCs with flexible parameters.

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