论文标题
关于最佳$（r，δ）$ lrcs的改进和构建的一些结果
Some Results on the Improved Bound and Construction of Optimal $(r,δ)$ LRCs
论文作者
论文摘要
Prakash \ emph {et al。}将$（r，δ）$ locality的本地维修代码（LRC）引入分布式存储系统（DSSS），因为它们的好处是通过同一本地组中的其他$ R $ Survival Nodes进行本地修复至少$δ-1 $擦除。实现$（r，δ）$ singleton型结合的LRC称为最佳$（R，δ）$ LRC。近年来，具有较长代码长度并确定最大代码长度的最佳$（R，δ）$ LRC的构造是编码理论的重要研究方向。在本文中，我们对最大$（R，δ）$ LRC的最大代码长度的改善进行了进一步的研究。对于$2δ+1 \ leq d \ leq2δ+2 $，我们的上限在很大程度上通过cai \ emph {et al。}改善了这些边界，在某些特殊情况下，它们很紧。此外，我们将Chen \ emph {等}的结果概括，并在有限的投影平面$ pg（2，q）$中获得最佳$（r = 2，δ）$ -LRC的完整表征。在这种几何表征中，我们基于向日葵结构构建了一类最佳$（R，δ）$ LRCS。结构和上限都比以前的界限更好。
Locally repairable codes (LRCs) with $(r,δ)$ locality were introduced by Prakash \emph{et al.} into distributed storage systems (DSSs) due to their benefit of locally repairing at least $δ-1$ erasures via other $r$ survival nodes among the same local group. An LRC achieving the $(r,δ)$ Singleton-type bound is called an optimal $(r,δ)$ LRC. Constructions of optimal $(r,δ)$ LRCs with longer code length and determining the maximal code length have been an important research direction in coding theory in recent years. In this paper, we conduct further research on the improvement of maximum code length of optimal $(r,δ)$ LRCs. For $2δ+1\leq d\leq 2δ+2$, our upper bounds largely improve the ones by Cai \emph{et al.}, which are tight in some special cases. Moreover, we generalize the results of Chen \emph{et al.} and obtain a complete characterization of optimal $(r=2, δ)$-LRCs in the sense of geometrical existence in the finite projective plane $PG(2,q)$. Within this geometrical characterization, we construct a class of optimal $(r,δ)$ LRCs based on the sunflower structure. Both the construction and upper bounds are better than previous ones.
