论文标题
最佳循环（R，δ）的新构建局部可从其零局部修复代码
New Constructions of Optimal Cyclic (r,δ) Locally Repairable Codes from Their Zeros
论文作者
论文摘要
Prakash等人引入了$（r，δ）$ - 本地维修代码（$（R，δ）$ -LRC）。 \ cite {prakash2012}用于耐受分布式存储系统中多个失败的节点，这是Gopalan等人产生的$ r $ -lrcs概念的概括。 \ cite {gopalan2012}。 $（r，δ）$ -LRC如果达到单例式结合，则据说它是最佳的。最近，Chen等。 \ cite {chen2018}概括了Tamo等人提出的环状$ r $ -lrcs的构建。 \ cite {tamo2015，tamo2016}，并构建了几类最佳$（r，δ）$ - lrcs长度为$ n $ for $ n \，| \，| \，（q-1）$或$ n \，（q-n \，| \，（q+1）$，分别以zeros Control的zeros Control和zeros Control的范围，分别为zeros Control和zeros Control。在\ cite {chen2018，chen2019}的工作之后，本文首先表征了$（r，δ）$ - 通过其零零的循环代码的局部性。然后，我们分别从两组零的产品中构造了几类最佳环状$（r，δ）$ - lrcs $ n $ for $ n \，| \，（q-1）$或$ n \，| \，（q+1）$。我们的构造包括所有最佳环状$（R，δ）$ -LRCS在\ cite {Chen2018，Chen2019}中提出的，我们的方法似乎更方便获得具有灵活参数的最佳循环$（R，δ）$ -LRC。此外，许多最佳的环状$（r，δ）$ - lrcs的长度$ n $ for $ n \，| \，（q-1）$或$ n \，| \，（q+1）$，因此可以从我们的方法中获得$（r+Δ1）\ nmid n $。
An $(r, δ)$-locally repairable code ($(r, δ)$-LRC for short) was introduced by Prakash et al. \cite{Prakash2012} for tolerating multiple failed nodes in distributed storage systems, which was a generalization of the concept of $r$-LRCs produced by Gopalan et al. \cite{Gopalan2012}. An $(r, δ)$-LRC is said to be optimal if it achieves the Singleton-like bound. Recently, Chen et al. \cite{Chen2018} generalized the construction of cyclic $r$-LRCs proposed by Tamo et al. \cite{Tamo2015,Tamo2016} and constructed several classes of optimal $(r, δ)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively in terms of a union of the set of zeros controlling the minimum distance and the set of zeros ensuring the locality. Following the work of \cite{Chen2018,Chen2019}, this paper first characterizes $(r, δ)$-locality of a cyclic code via its zeros. Then we construct several classes of optimal cyclic $(r, δ)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively from the product of two sets of zeros. Our constructions include all optimal cyclic $(r,δ)$-LRCs proposed in \cite{Chen2018,Chen2019}, and our method seems more convenient to obtain optimal cyclic $(r, δ)$-LRCs with flexible parameters. Moreover, many optimal cyclic $(r,δ)$-LRCs of length $n$ for $n\, |\, (q-1)$ or $n\,|\, (q+1)$, respectively such that $(r+δ-1)\nmid n$ can be obtained from our method.
