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

TL;DR

This paper introduces a family of binary locally repairable codes that enable sequential repair of multiple erasures, demonstrating significantly improved repair capabilities over parallel approaches in distributed storage systems.

Contribution

The paper constructs a new class of binary codes equivalent to the direct product of single-parity-check codes, allowing sequential repair of up to 2^m-1 erasures, which surpasses previous parallel repair limits.

Findings

Codes permit sequential repair for up to 2^m-1 erasures

Sequential approach outperforms parallel repair in efficiency

Constructed codes are based on direct product of single-parity-check codes

Abstract

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.