Capacity of Locally Recoverable Codes

TL;DR

This paper investigates the performance of locally recoverable codes (LRCs) in channels with random erasures or errors, providing a tight capacity gap characterization for input-symmetric channels.

Contribution

It offers the first analysis of LRCs' performance on stochastic channels, extending understanding beyond their traditional applications in distributed storage.

Findings

Tight capacity gap characterization for LRCs on input-symmetric channels

Analysis applicable to LRCs correcting multiple local erasures

Results bridge the gap between coding theory and channel capacity understanding

Abstract

Motivated by applications in distributed storage, the notion of a locally recoverable code (LRC) was introduced a few years back. In an LRC, any coordinate of a codeword is recoverable by accessing only a small number of other coordinates. While different properties of LRCs have been well-studied, their performance on channels with random erasures or errors has been mostly unexplored. In this paper, we analyze the performance of LRCs over such stochastic channels. In particular, for input-symmetric discrete memoryless channels, we give a tight characterization of the gap to Shannon capacity when LRCs are used over the channel. Our results hold for a general notion of LRCs that correct multiple local erasures.