Quantum $(r,\delta)$-locally recoverable codes

A classical $(r,\delta)$-locally recoverable code is an error-correcting code such that, for each coordinate $c_i$ of a codeword, there exists a set of at most $r+ \delta -1$ coordinates containing $c_i$ which allow us to correct any $\delta -1$ erasures in that set. These codes are useful for avoiding loss of information in large scale distributed and cloud storage systems. In this paper, we introduce quantum $(r,\delta)$-locally recoverable codes and give a necessary and sufficient condition for a quantum stabilizer code $Q(C)$ to be $(r,\delta)$-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 a quantum stabilizer code 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 $(r,\delta)$-local recoverability. A Singleton-like bound is stated in this case and examples of optimal stabilizer $(r,\delta)$-locally recoverable codes are given.