互联网和大数据催生了数据中心的迅猛发展，在数据中心等分布式存储系统中，存储节点失效经常发生，迫切需要冗余备份和快速修复等核心技术，局部修复码(LRC)作为一种新的分布式存储冗余策略已成为近年来编码信息论领域的一大研究热点。以往的LRC研究主要是基于线性码生成矩阵进行的，集中于极小距离理论界和最优码构造两方面。本课题利用经典编码理论和LDPC码中的技巧和方法，从校验矩阵角度研究LRC理论与应用，主要包括：LRC的校验矩阵刻画、极小距离理论界、最优或循环LRC的构造、多组或不等局部修复模型、LRC的结构性质以及分布式存储应用，等等。本课题试图在新的校验矩阵研究框架下，首先将已有的重要结果统一起来并解释其联系，然后对于循环、小的有限域、多修复组、新局部修复模型等重要情形，给出LRC新的理论界或最优构造，并讨论其结构性质及应用。这些工作对于丰富深化LRC基本理论并推进其实际应用具有重要意义。

Internet and big data accelerate the rapid development of data centers. In distributed storage systems of data centers, storage nodes fail frequently, which implies that data redundancy and rapid recovery are necessary. As a new data redundancy strategy in distributed storage systems, locally repairable codes (LRCs) become a research focus in the areas of coding and information theory recently. The previous studies on an LRC are mainly performed through its generator matrix, focus on two aspects: theoretic minimum distance bounds and optimal LRC constructions. In this project, we employ skills and methods from classical coding theory and LDPC codes, study the LRC theory and applications by a parity-check matrix approach. The main contents of the project include the parity-check matrix characterization of an LRC, minimum distance bounds, constructions of optimal or cyclic LRCs, multiple locality or unequal locality models, the structure properties of LRCs and their applications in distributed storage systems, etc. Under the research framework of parity-check matrix, this project tries to unify the previous important results and explains their connections. Then for important cases of LRCs, e.g. cyclic, small finite field, multiple repair groups, new local repair models, etc., we will obtain new theoretic bounds or optimal constructions, and discuss the structure properties of optimal LRCs and their application in distributed storage systems. These works are very important for the basic theory and practical applications of LRCs.