Bounds and Constructions of Codes with Multiple Localities

TL;DR

This paper investigates bounds and constructions for locally repairable codes with multiple localities, extending existing bounds and providing optimal code constructions for such codes.

Contribution

It extends the Singleton and alphabet-dependent bounds to ML-LRCs with more than two localities and constructs optimal codes achieving these bounds.

Findings

Extended bounds for ML-LRCs with multiple localities.

Constructed Singleton-optimal ML-LRCs.

Developed a family of binary ML-LRCs based on generalized code concatenation.

Abstract

This paper studies bounds and constructions of locally repairable codes (LRCs) with multiple localities so-called multiple-locality LRCs (ML-LRCs). In the simplest case of two localities some code symbols of an ML-LRC have a certain locality while the remaining code symbols have another one. We extend two bounds, the Singleton and the alphabet-dependent upper bound on the dimension of Cadambe--Mazumdar for LRCs, to the case of ML-LRCs with more than two localities. Furthermore, we construct Singleton-optimal ML-LRCs as well as codes that achieve the extended alphabet-dependent bound. We give a family of binary ML-LRCs based on generalized code concatenation that is optimal with respect to the alphabet-dependent bound.