论文标题
超级线性长度的最佳本地维修代码的新结构
New Constructions of Optimal Locally Repairable Codes with Super-Linear Length
论文作者
论文摘要
作为现代分布式存储系统中的重要编码方案，本地维修代码（LRC）从实际应用和理论研究的角度吸引了很多关注。作为LRCS研究的主要主题，相应最佳代码的界限和构造特别关注。 在这项工作中，具有$（r，δ）$的代码 - 具有最佳距离W.R.T.的局部性Prakash等人给出的约束。 \ cite {prakash2012optimal}被考虑。通过均等检查矩阵方法，提供了所有符号局部性（$（r，δ）_a $ -lrcs）和最佳$（r，δ）$ -LRCS的最佳$（R，δ）$ -LRCS的构造，提供了信息局部遗址（$（r，δ）_i _i $ -lrcs）。作为Xing和yuan \ cite {xy19}的作品的概括，这些结构建立在稀疏超图和最佳$（R，δ）$ -LRC之间的连接上。借助大型稀疏超图的结构，构造的代码长度可以在字母大小上是超级线性。当代码的最小距离至少为$3δ+1 $时，这会改善以前的结构。作为两个应用，还构建了具有超级线性长度和无界长度的GSD代码的最佳H-LRC。
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. \cite{Prakash2012Optimal} 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 \cite{XY19}, 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.
