Optimal Locally Repairable Codes and Connections to Matroid Theory

TL;DR

This paper introduces explicit, optimal locally repairable codes (LRCs) for distributed storage, leveraging Reed-Solomon codes and matroid theory to improve repair efficiency and approach fundamental bounds.

Contribution

It presents a simple, explicit construction of optimal LRCs based on Reed-Solomon codes and matroid analysis, filling a gap in existing code designs.

Findings

Constructed explicit optimal LRCs with low repair locality.

Derived a new matroid-based result for code generator matrices.

Demonstrated the practical relevance of LRCs in large-scale storage systems.

Abstract

Petabyte-scale distributed storage systems are currently transitioning to erasure codes to achieve higher storage efficiency. Classical codes like Reed-Solomon are highly sub-optimal for distributed environments due to their high overhead in single-failure events. Locally Repairable Codes (LRCs) form a new family of codes that are repair efficient. In particular, LRCs minimize the number of nodes participating in single node repairs during which they generate small network traffic. Two large-scale distributed storage systems have already implemented different types of LRCs: Windows Azure Storage and the Hadoop Distributed File System RAID used by Facebook. The fundamental bounds for LRCs, namely the best possible distance for a given code locality, were recently discovered, but few explicit constructions exist. In this work, we present an explicit and optimal LRCs that are simple to construct. Our construction is based on grouping Reed-Solomon (RS) coded symbols to obtain RS coded symbols over a larger finite field. We then partition these RS symbols in small groups, and re-encode them using a simple local code that offers low repair locality. For the analysis of the optimality of the code, we derive a new result on the matroid represented by the code generator matrix.