Bounds on Binary Locally Repairable Codes Tolerating Multiple Erasures

TL;DR

This paper investigates binary locally repairable codes capable of handling multiple erasures, establishing bounds on their minimum distance and providing examples that meet these bounds, with tools from matroid theory.

Contribution

It introduces bounds on the minimum distance of binary LRCs with multiple erasures and demonstrates the atomic structure of binary matroids related to these codes.

Findings

Derived bounds on minimum distance for binary LRCs

Constructed examples achieving these bounds

Showed the atomic nature of the lattice of cyclic flats in binary matroids

Abstract

Recently, locally repairable codes has gained significant interest for their potential applications in distributed storage systems. However, most constructions in existence are over fields with size that grows with the number of servers, which makes the systems computationally expensive and difficult to maintain. Here, we study linear locally repairable codes over the binary field, tolerating multiple local erasures. We derive bounds on the minimum distance on such codes, and give examples of LRCs achieving these bounds. Our main technical tools come from matroid theory, and as a byproduct of our proofs, we show that the lattice of cyclic flats of a simple binary matroid is atomic.