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

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.