New Constructions of Optimal Locally Repairable Codes with Super-Linear Length

TL;DR

This paper introduces new methods for constructing optimal locally repairable codes with super-linear length, leveraging sparse hypergraph connections, thus advancing code efficiency in distributed storage systems.

Contribution

The work provides novel constructions of optimal $(r, ext{delta})$-LRCs with super-linear length using hypergraph-based approaches, improving previous bounds for codes with minimum distance at least $3 ext{delta}+1$.

Findings

Codes with super-linear length are achievable.

Constructions improve bounds for minimum distance ≥ 3delta+1.

Applications include optimal H-LRCs and GSD codes with unbounded length.

Abstract

As an important coding scheme in modern distributed storage systems, locally repairable codes (LRCs) have attracted a lot of attentions from perspectives of both practical applications and theoretical research. As a major topic in the research of LRCs, bounds and constructions of the corresponding optimal codes are of particular concerns. In this work, codes with $(r,\delta)$-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. \cite{Prakash2012Optimal} are considered. Through parity check matrix approach, constructions of both optimal $(r,\delta)$-LRCs with all symbol locality ($(r,\delta)_a$-LRCs) and optimal $(r,\delta)$-LRCs with information locality ($(r,\delta)_i$-LRCs) are provided. As a generalization of a work of Xing and Yuan \cite{XY19}, these constructions are built on a connection between sparse hypergraphs and optimal $(r,\delta)$-LRCs. With the help of constructions of large sparse hypergraphs, the length of codes constructed can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least $3\delta+1$. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed.