Optimal quantum 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 (quantum LRCs) are the quantum counterpart of classical LRCs. They allow us to correct erasures at several positions from a trace-preserving quantum operation acting on qudits of a larger set of positions. Parameters and localities of quantum LRCs satisfy a Singleton-like bound; codes attaching this bound are named to be optimal. Quantum LRCs, $\mathcal{Q}(\mathcal{C})$, can be constructed from classical Hermitian (or Euclidean) dual containing codes $\mathcal{C}$, and their recovery abilities are upper bounded by the minimum distance of the Hermitian (or Euclidean) dual of those codes. We consider matrix-product codes (MPCs) $\mathcal{C}$ and give constituent matrices and conditions on the constituent codes such that the codes $\mathcal{C}$ satisfy the conditions to provide quantum LRCs. As consequence, we are able to provide the locality and parameters of the quantum LRCs $\mathcal{Q}(\mathcal{C})$ and determine families of optimal quantum LRCs derived from them.