New Constructions of Optimal Locally Repairable Codes with Super-Linear Length
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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. <cit.> are considered. Through parity check matrix approach, constructions of both optimal (r,δ)-LRCs with all symbol locality ((r,δ)_a-LRCs) and optimal (r,δ)-LRCs with information locality ((r,δ)_i-LRCs) are provided. As a generalization of a work of Xing and Yuan <cit.>, 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 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 3δ+1. As two applications, optimal H-LRCs with super-linear length and GSD codes with unbounded length are also constructed.
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