New Constructions of Optimal Locally Repairable Codes With Super-Linear 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,δ)-locality which have optimal minimal distance w.r.t. the bound given by Prakash et al. are considered. Through parity-check matrix approach, constructions of both optimal (r,δ)-LRCs with all symbol locality ( (r,δ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</sub> -LRCs) and optimal (r,δ)-LRCs with information locality ( (r,δ) <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> -LRCs) are provided. As a generalization of a work of Xing and Yuan, these constructions are built on a connection between sparse hypergraphs and optimal (r,δ)-LRCs. With the help of constructions of large sparse hypergraphs, the lengths of codes obtained from our construction can be super-linear in the alphabet size. This improves upon previous constructions when the minimal distance of the code is at least 3δ+1. As two applications, optimal H-LRCs with super-linear lengths and GSD codes with unbounded lengths are also constructed.