Optimal Quantum $(r,\delta)$-Locally Repairable Codes From Matrix-Product Codes

This paper studies optimal quantum $(r,\delta)$-LRCs from matrix-product (MP) codes. We establish a necessary and sufficient condition for an MP code to be an optimal $(r,\delta)$-LRC. Based on this, we present a characterization for optimal quantum $(r,\delta)$-LRCs from MP codes with nested constituent codes, and also study optimal quantum $(r,\delta)$-LRCs constructed from MP codes with non-nested constituent codes. Through Hermitian dual-containing and Euclidean dual-containing MP codes, we present five infinite families of optimal quantum $(r,\delta)$-LRCs with flexible parameters.