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Binary Locally Repairable Codes ---Sequential Repair for Multiple Erasures

Locally repairable codes (LRC) for distribute storage allow two approaches to locally repair multiple failed nodes: 1) parallel approach, by which each newcomer access a set of $r$ live nodes $(r$ is the repair locality$)$ to download data and recover the lost packet; and 2) sequential approach, by which the newcomers are properly ordered and each newcomer access a set of $r$ other nodes, which can be either a live node or a newcomer ordered before it. An $[n,k]$ linear code with locality $r$ and allows local repair for up to $t$ failed nodes by sequential approach is called an $(n,k,r,t)$-exact locally repairable code (ELRC). In this paper, we present a family of binary codes which is equivalent to the direct product of $m$ copies of the $[r+1,r]$ single-parity-check code. We prove that such codes are $(n,k,r,t)$-ELRC with $n=(r+1)^m,k=r^m$ and $t=2^m-1$, which implies that they permit local repair for up to $2^m-1$ erasures by sequential approach. Our result shows that the sequential approach has much bigger advantage than parallel approach.