A locally repairable code (LRC) is a linear code such that every code symbol can be recovered by accessing a small number of other code symbols. In this paper, we study bounds and constructions of LRCs from the viewpoint of parity-check matrices. Firstly, a simple and unified framework based on parity-check matrix to analyze the bounds of LRCs is proposed, and several new explicit bounds on the minimum distance of LRCs in terms of the field size are presented. In particular, we give an alternate proof of the Singleton-like bound for LRCs first proved by Gopalan et al. Some structural properties on optimal LRCs that achieve the Singleton-like bound are given. Then, we focus on constructions of optimal LRCs over the binary field. It is proved that there are only five classes of possible parameters with which optimal binary LRCs exist. Moreover, by employing the proposed parity-check matrix approach, we completely enumerate all these five classes of optimal binary LRCs attaining the Singleton-like bound in the sense of equivalence of linear codes.

中文翻译：

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局部可修复码的边界和构造：奇偶校验矩阵方法

本地可修复代码 (LRC) 是一种线性代码，因此可以通过访问少量其他代码符号来恢复每个代码符号。在本文中，我们从奇偶校验矩阵的角度研究 LRC 的边界和构造。首先，提出了一个基于奇偶校验矩阵来分析LRCs边界的简单统一的框架，并提出了几个新的LRCs最小距离在字段大小方面的明确边界。特别是，我们给出了 Gopalan 等人首先证明的 LRC 类单例边界的替代证明。给出了实现类单例边界的最佳 LRC 的一些结构特性。然后，我们专注于在二元域上构建最佳 LRC。事实证明，只有五类可能的参数与最优二元 LRC 存在。而且，