论文标题
最佳的第四纪本地可修复的代码，达到了单身型绑定
Optimal Quaternary Locally Repairable Codes Attaining the Singleton-like Bound
论文作者
论文摘要
近年来，引入了几种新类型的代码，以提供分布式存储系统中的耐故障和保证系统可靠性，其中本地维修代码（简称LRC）发挥了重要作用。 如果可以通过访问最多$ r $其他代码符号来修复其每个代码符号，则据说线性代码具有局部性$ r $。对于具有长度$ n $的LRC，尺寸$ k $和local $ r $，其最小距离$ d $被证明满足了单身型绑定的$ d \ leq n-k- \ lceil k/r \ r \ rceil+2 $。从那以后，已经完成了许多工作，用于构建LRCS在小田地上遇到类似单元的界限。 在本文中，我们研究了第四纪LRCS通过平等检查矩阵方法遇到的类似单例的遇到。使用有限几何形状的工具，我们为LRC提供了一些最佳的新必要条件。从中，我们证明了最佳第四纪LRC的$ 27 $不同类别的参数。此外，对于每个类，都会介绍相应最佳四级LRC的显式结构。
Recent years, several new types of codes were introduced to provide fault-tolerance and guarantee system reliability in distributed storage systems, among which locally repairable codes (LRCs for short) have played an important role. A linear code is said to have locality $r$ if each of its code symbols can be repaired by accessing at most $r$ other code symbols. For an LRC with length $n$, dimension $k$ and locality $r$, its minimum distance $d$ was proved to satisfy the Singleton-like bound $d\leq n-k-\lceil k/r\rceil+2$. Since then, many works have been done for constructing LRCs meeting the Singleton-like bound over small fields. In this paper, we study quaternary LRCs meeting Singleton-like bound through a parity-check matrix approach. Using tools from finite geometry, we provide some new necessary conditions for LRCs being optimal. From this, we prove that there are $27$ different classes of parameters for optimal quaternary LRCs. Moreover, for each class, explicit constructions of corresponding optimal quaternary LRCs are presented.
