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Quantum $(r,δ)$-locally recoverable codes

Classical $(r,δ)$-locally recoverable codes are designed for avoiding loss of information in large scale distributed and cloud storage systems. We introduce the quantum counterpart of those codes by defining quantum $(r,δ)$-locally recoverable codes which are quantum error-correcting codes capable of correcting $δ-1$ qudit erasures from sets of at most $r+ δ-1$ qudits. We give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,δ)$-locally recoverable. Our condition depends only on the puncturing and shortening at suitable sets of both the symplectic self-orthogonal code $C$ used for constructing $Q(C)$ and its symplectic dual $C^{\perp_s}$. When $Q(C)$ comes from a Hermitian or Euclidean dual-containing code, and under an extra condition, we show that there is an equivalence between the classical and quantum concepts of $(r,δ)$-local recoverability. A Singleton-like bound is stated in this case and examples attaining the bound are given.