Entanglement-assisted quantum locally recoverable codes: bounds and constructions with availability
In this work, we define entanglement-assisted quantum locally recoverable codes with availability, in which any set of up to $\delta-1$ erased qudits can be recovered from any one of $t$ local recovery sets, each of size at most $r+\delta-1$, with the recovery sets intersecting exactly in the erased coordinates, where $r$ is a (small) positive integer. We show that shared entanglement permits $t>1$, meaning that multiple local recovery sets can be available for the same set of up to $\delta-1$ erasures. We establish a Singleton-like bound for this family of codes and present random constructions based on classical linear codes with Vandermonde parity-check matrices. We also provide explicit constructions of entanglement-assisted quantum locally recoverable codes with availability from several classical code families and their folded versions, including Tamo-Barg codes, fiber-product codes, and algebraic-geometry codes such as one-point Hermitian and Suzuki codes.