Finite Fields and Their Applications ( IF 1.2 ) Pub Date : 2020-01-29 , DOI: 10.1016/j.ffa.2020.101650 Weijun Fang , Fang-Wei Fu
Locally repairable codes with locality r (r-LRCs for short) were introduced by Gopalan et al. [1] to recover a failed node of the code from at most other r available nodes. And then -locally repairable codes (-LRCs for short) were produced by Prakash et al. [2] for tolerating multiple failed nodes. An r-LRC can be viewed as an -LRC. An -LRC is called optimal if it achieves the Singleton-type bound. It has been a great challenge to construct q-ary optimal -LRCs with length much larger than q. Surprisingly, Luo et al. [3] presented a construction of q-ary optimal r-LRCs of minimum distances 3 and 4 with unbounded lengths (i.e., lengths of these codes are independent of q) via cyclic codes.
In this paper, inspired by the work of [3], we firstly construct two classes of optimal cyclic -LRCs with unbounded lengths and minimum distances or , which generalize the results about the case given in [3]. Secondly, with a slightly stronger condition, we present a construction of optimal cyclic -LRCs with unbounded length and larger minimum distance 2δ. Furthermore, when , we give another class of optimal cyclic -LRCs with unbounded length and minimum distance 6.
中文翻译:
具有无限长度的最佳循环(r,δ)局部可修复码
Gopalan等人介绍了局部性为r的局部可修复代码(r -LRCs)。[1]从最多其他r个可用节点中恢复代码的失败节点。接着-本地可修复代码(-LRCs)由Prakash等人生产。[2]用于容忍多个故障节点。一个ř -LRC可以被看作是一个-LRC。一个如果达到单例类型界限,则-LRC被称为最优。构造q元最优值一直是一个巨大的挑战长度远大于q的-LRC 。令人惊讶的是,Luo等。文献[3]通过循环码提出了最小距离为3和4的q元最优r -LRC的构造,其无限长度(即,这些码的长度与q无关)。
在本文中,受[3]的启发,我们首先构造了两类最优循环 -LRC具有无限制的长度和最小距离 要么 ,将关于 在[3]中给出的情况。其次,在条件稍强的情况下,我们提出了最优周期的构造-LRCs与无界的长度和较大的最小距离2 δ。此外,当,我们给出了另一类最优循环 -无限制长度和最小距离的LRC 6。