Optimal pure quantum $(r,δ)$-locally recoverable codes from matrix-product construction
Locally recoverable codes (LRCs) are classical error-correcting codes widely used in large-scale distributed and cloud storage systems. Quantum locally recoverable codes of locality $(r,δ)$ (quantum $(r,δ)$-LRCs) are the quantum counterpart of classical $(r,δ)$-LRCs. They allow us to correct erasures at several positions using a trace-preserving quantum operation acting on qudits of a larger set of positions. Quantum $(r,δ)$-LRCs, $\mathcal{Q}(\mathcal{C})$, can be constructed from classical Euclidean (or Hermitian) dual-containing codes $\mathcal{C}$, and their recovery abilities are upper bounded by the minimum distance of the Euclidean (or Hermitian) dual of those codes. Parameters and localities of pure quantum $(r,δ)$-LRCs satisfy a Singleton-like bound; codes attaining equality are referred to as optimal. We consider matrix-product codes (MPCs) $\mathcal{C}$ and give constituent (or defining) matrices and conditions on the constituent codes such that the codes $\mathcal{C}$ satisfy the conditions to provide quantum $(r,δ)$-LRCs. As a consequence, we are able to determine their locality and parameters. Furthermore, we determine families of optimal pure quantum $(r,δ)$-LRCs derived from them.