A Class of Codes With Availability and Multiple Local-Erasures Correction

In a locally recoverable code (LRC), any code symbol can be recovered by accessing at most $r$ other symbols (called a recovery set). In an LRC with availability , any information symbol has $t$ disjoint recovery sets. In this letter, we consider a new class of codes with availability, where the $l^{th}, 1 \leq l \leq t$ disjoint recovery set for any information symbol has locality $r_{l}$ and it is protected by a local code of minimum Hamming distance at least $\delta _{l}$ . We derive an upper-bound on the minimum Hamming distance of these codes. A family of systematic codes with information availability is constructed achieving the bound with equality.