Locally Repairable Codes

Abstract

Distributed storage systems for large-scale applications typically use replication for reliability. Recently, erasure codes were used to reduce the large storage overhead, while increasing data reliability. A main limitation of off-the-shelf erasure codes is their high-repair cost during single node failure events. A major open problem in this area has been the design of codes that i) are repair efficient and ii) achieve arbitrarily high data rates.In this paper, we explore the repair metric of locality, which corresponds to the number of disk accesses required during a single node repair. Under this metric we characterize an information theoretic trade-off that binds together locality, code distance, and the storage capacity of each node. We show the existence of optimal locally repairable codes (LRCs) that achieve this trade-off. The achievability proof uses a locality aware flowgraph gadget which leads to a randomized code construction. Finally, we present an optimal and explicit LRC that achieves arbitrarily high data-rates. Our locality optimal construction is based on simple combinations of Reed-Solomon blocks.Parts of this work were presented in [1].

Partial MDS (PMDS) and Sector-Disk (SD) codes that tolerate the erasure of two random sectors

“…In fact, (1; s) PMDS, called Maximally Recoverable codes in [6], satisfy also the requirements of Locally Repairable codes [10], [13]. Additionally, the recently-defined STAIR codes relax the failure-coverage of SD codes in order to allow for general constructions [8].…”

Generalized construction of cyclic (<i>r</i>, <i>t</i>)-locally repairable codes using trace function

“…Locally repairable codes (LRCs) [1,2] have recently attracted considerable attention as a promising technology for data protection. An ðr; tÞ-LRC is a code having the following property: each coordinate of codewords can be recovered from at most r other coordinates (called a repair set), and there are at least t disjoint repair sets for each coordinate, where r and t are called locality and availability, respectively [3].…”

Optimal locally repairable codes with local minimum storage regeneration via rank-metric codes

“…(Han and Lastras-Montano [18] provide a similar upper bound which is coincident with the one in [11] for small minimum distances, and also present codes that attain this bound in the context of reliable memories.) In [10], Papailiopoulos and Dimakis generalize the bound in [9] to vector codes, and present locally repairable coding schemes which exhibits MDS property at the cost of small amount of additional storage per node.…”

“…Subsequently, in Section III, we derive an upper bound on the minimum distance d min of the vector codes that satisfy a given locality constraint, which establishes a trade off between node failure resilience (i.e., d min ) and per node storage α. 1 The bound presented in [10] can be considered as a special case of our bound with δ = 2. Further, we present an explicit construction for LRCs which attain this bound on minimum distance.…”