On the Average Locality of Locally Repairable Codes

Abstract

A linear block code with dimension $k$ , length $n$ , and minimum distance $d$ is called a locally repairable code (LRC) with locality $r$ , if it can retrieve any coded symbol by at most $r$ other coded symbols. LRCs have been recently proposed and used in practice in distributed storage systems, such as Windows Azure Storage and Facebook HDFS-RAID. Theoretical bounds on the maximum locality of LRCs ( $r$ ) have been established. The average locality of an LRC ( $\overline {r}$ ) directly affects the costly repair bandwidth, disk I/O, and the number of nodes involved in the repair process of a missing data block. There is a gap in the literature studying $\overline {r}$ . In this paper, we establish a lower bound on $\overline {r}$ of arbitrary $(n,k,d)$ LRCs. Furthermore, we obtain a tight lower bound on $\overline {r}$ for a practical case where the code rate $(R=({k}/{n}))$ is greater than $(1-({1}/{\sqrt {n}}))^{2}$ . Finally, we design three classes of LRCs that achieve the obtained bounds on $\overline {r}$ . Comparing with the existing LRCs, our proposed codes improve the average locality without sacrificing such crucial parameters as the code rate or minimum distance.