Binary Locally Repairable Codes - Sequential Repair for Multiple Erasures

Locally repairable codes (LRC) for distributed 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 that 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] singleparity-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.