论文标题
与小区域的单身最佳的本地可修复代码的界限和结构
Bounds and Constructions of Singleton-Optimal Locally Repairable Codes with Small Localities
论文作者
论文摘要
近年来，已经对实现单身型结合的最佳本地维修代码（LRC）的结构进行了详尽的研究。在本文中，我们考虑了Minmum距离$ d = 6 $，local $ r = 3 $和最小距离$ d = 7 $和local $ r = 2 $的新界和构造的新界限和构造。首先，我们在这两个LRC家族的存在与具有某些属性的投影空间中的某些线集存在之间建立了等效的联系。然后，我们采用线点的入射率矩阵和约翰逊的界限来实现恒定重量代码，从而在代码长度上得出了新的改进界限，这比已知的结果更紧。最后，通过使用一些有限磁场和有限几何形状的技术，我们提供了一些新构造的单胎最佳LRC，它们的长度比以前的长度更大。
Constructions of optimal locally repairable codes (LRCs) achieving Singleton-type bound have been exhaustively investigated in recent years. In this paper, we consider new bounds and constructions of Singleton-optimal LRCs with minmum distance $d=6$, locality $r=3$ and minimum distance $d=7$ and locality $r=2$, respectively. Firstly, we establish equivalent connections between the existence of these two families of LRCs and the existence of some subsets of lines in the projective space with certain properties. Then, we employ the line-point incidence matrix and Johnson bounds for constant weight codes to derive new improved bounds on the code length, which are tighter than known results. Finally, by using some techniques of finite field and finite geometry, we give some new constructions of Singleton-optimal LRCs, which have larger length than previous ones.
