Title: Locally Repairable Codes with Multiple $(r_{i}, δ_{i})$-Localities,
Abstract: In distributed storage systems, locally repairable codes (LRCs) are
introduced to realize low disk I/O and repair cost. In order to tolerate
multiple node failures, the LRCs with \emph{$(r, \delta)$-locality} are further
proposed. Since hot data is not uncommon in a distributed storage system, both
Zeh \emph{et al.} and Kadhe \emph{et al.} focus on the LRCs with \emph{multiple
localities or unequal localities} (ML-LRCs) recently, which said that the
localities among the code symbols can be different. ML-LRCs are attractive and
useful in reducing repair cost for hot data. In this paper, we generalize the
ML-LRCs to the $(r,\delta)$-locality case of multiple node failures, and define
an LRC with multiple $(r_{i}, \delta_{i})_{i\in [s]}$ localities ($s\ge 2$),
where $r_{1}\leq r_{2}\leq\dots\leq r_{s}$ and
$\delta_{1}\geq\delta_{2}\geq\dots\geq\delta_{s}\geq2$. Such codes ensure that
some hot data could be repaired more quickly and have better failure-tolerance
in certain cases because of relatively smaller $r_{i}$ and larger $\delta_{i}$.
Then, we derive a Singleton-like upper bound on the minimum distance for the
proposed LRCs by employing the regenerating-set technique. Finally, we obtain a
class of explicit and structured constructions of optimal ML-LRCs, and further
extend them to the cases of multiple $(r_{i}, \delta)_{i\in [s]}$ localities.