Codes with Availability and different Localities

Abstract

Any code symbol in a locally recoverable code (LRC) can be recovered by accessing at most r other code symbols (called a recovery set). Codes where any local code is protected by a local code of minimum Hamming distance at least δ is called an $(r,\delta)$ code. If each code symbol has t disjoint recovery sets then the code is called an LRC with availability. In this letter we consider a new class of codes with availability. In these codes the $l^{th}, 1\leq l\leq t$ disjoint recovery set for any code symbol has locality $r\iota$ and it will be protected by a local code of minimum Hamming distance at least $\delta_{l}$. We provide a new definition for availability. We derive an upper bound on the minimum Hamming distance of the new class of codes. We describe an additional property of this code that it allows all the k information symbols to be accessed t times simultaneously and thus providing availability t for all the k information symbols.