Repair locality has been an important metric in a distributed storage system (DSS). Erasure codes with small locality are more popular in a DSS, which means fewer available nodes participating in the repair process of failed nodes. Locally repairable codes (LRCs) as a new coding scheme have given more rise to the system performance and attracted a lot of interest in the theoretical research in coding theory. The particular concern among the research problems is the bounds and optimal constructions of LRCs. The problem of optimal constructions of LRCs includes the most important case of Singleton-optimal LRCs whose minimum distance achieves the Singleton-like bound, which is the core consideration in this paper. In this work, we first of all derive an improved and general upper bound on the code length of Singleton-optimal LRCs with minimum distance $d=5, 6$ , some known constructions are shown to exactly achieve our new bound, which verifies its tightness. For locality $r=2$ and distance $d=6$ , we construct three new Singleton-optimal LRCs whose code length $n=3(q+1)$ , $n=3(q+\sqrt {q}+1)$ and $n=3(2q-4)$ , respectively. Moreover, we obtain a complete characterization for Singleton-optimal LRCs with $r=2$ and $d=6$ . Such characterization has established an important connection between the existence of Singleton-optimal LRCs and that of a special subset of lines of finite projective plane $PG(2, q)$ , thus provides a methodology for constructing LRCs with longer length based on any advance on finite projective plane $PG(2, q)$ . In the end, we employ the well-known line-point incidence matrix and Johnson bounds for constant weight codes to derive tighter upper bounds on the code length. These new bounds further help us to prove that some of the previous Singleton-optimal constructions or their extensions achieve the longest possible code length for $q=3, 4, 5, 7$ . It’s worth noting that all of our Singleton-optimal constructions possess small locality $r=2$ , which are attractive in a DSS.

中文翻译：

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最小距离为 5 和 6 的局部可修复代码的改进边界和单例最优构造

修复位置一直是分布式存储系统 (DSS) 中的一个重要指标。具有小局部性的纠删码在 DSS 中更受欢迎，这意味着参与故障节点修复过程的可用节点更少。局部可修复码（LRCs）作为一种新的编码方案，极大地提升了系统性能，引起了编码理论理论研究的极大兴趣。研究问题中特别关注的是 LRC 的边界和最优构造。LRCs 的优化构造问题包括最重要的单例最优 LRCs 的情况，其最小距离达到类单例边界，这是本文的核心考虑。在这项工作中， $d=5, 6$ ，一些已知的结构被证明可以准确地实现我们的新边界，这验证了它的紧密性。对于地区 $r=2$ 和距离 $d=6$ ，我们构造了三个新的单例最优 LRC，其代码长度 $n=3(q+1)$ , $n=3(q+\sqrt {q}+1)$ 和 $n=3(2q-4)$ ， 分别。此外，我们获得了单例最优 LRC 的完整表征 $r=2$ 和 $d=6$ . 这种表征在单例最优 LRC 的存在与有限投影平面线的特殊子集的存在之间建立了重要联系 $PG(2, q)$ ，因此提供了一种基于有限投影平面上的任何进展构造具有更长长度的 LRC 的方法 $PG(2, q)$ . 最后，我们使用众所周知的线点关联矩阵和约翰逊边界来计算恒定权重代码，以推导出更严格的代码长度上限。这些新的界限进一步帮助我们证明之前的一些单例最优结构或其扩展实现了最长的代码长度 $q=3, 4, 5, 7$ . 值得注意的是，我们所有的单例最优结构都具有小局部性 $r=2$ ，这在 DSS 中很有吸引力。