达到Singleton类型界的最优局部修复码研究

Recently, with the rapid development of Internet, locally repairable codes (LRCs) have attracted a lot of interest as a new data redundancy strategy for distributed storage systems in the areas of coding and information theory. Singleton-like bounds are very important bounds for the minimum distances of LRCs, which can be seen as generalizations of classical Singleton bound when considering the locality constraints. The optimal code construction and determination of the maximal code length of q-ary maximum distance separable (MDS) codes (e.g., MDS Main Conjecture) are very important research problems in coding theory. Optimal LRCs achieving the Singleton-like bounds are also of great theoretical and practical significance. In this project, we employ skills and methods from classical coding theory to study optimal LRCs meeting the Singleton-like bounds by using a parity-check matrix approach. Firstly, this project provides a new view to understand the Singleton-like bounds and explores the connections between optimal LRCs and MDS codes; Then, this project tries to give optimal constructions of several kinds of optimal LRCs meeting the Singleton-like bounds over small finite fields, determine the maximal code length of optimal q-ary LRCs, study the structural properties and constructions of LRCs with maximal code lengths, etc.. These works are very important for both the basic theory and practical applications of optimal LRCs.

随着互联网的快速发展，近年来局部修复码(LRC)作为一种新的分布式存储冗余策略在编码信息论领域引起极大关注。Singleton类型界为LRC最重要的极小距离理论界，可视为经典Singleton界添加局部性后的推广。达到Singleton界的最大距离可分(MDS)码的最优构造，最大码长，如“MDS码主猜想”等为编码领域重要研究问题，达到Singleton类型界的最优LRC在理论和应用上同样具有重大意义。本课题利用经典编码理论的技巧和方法，从校验矩阵角度研究达到Singleton类型界的最优局部修复码，首先对已有重要理论界给出新的解释并探讨最优LRC与MDS码的联系；然后构造小域上达到Singleton类型界的最优LRC、确定给定有限域上最优LRC的最大码长、研究具有最大码长的LRC的构造、性质以及分布式存储应用，等等。这些工作对于进一步深化最优LRC的基本理论并推进其应用具有重要意义。

局部修复码（LRC）作为数据容错编码在分布式存储系统中具有重要应用，本项目从校验矩阵角度对达到Singleton类型界的最优LRC的编码构造、最大码长和结构性质等问题进行了深入研究。在编码构造方面，我们从理论上确定了2元和4元域上最优(n,k,r)-LRC可取得的全部参数，并通过给出校验矩阵得到了全部参数下的最优构造，同时给出了3元域上最优(n,k,r,δ)-LRC的全部可取得的参数和相应的最优构造；最大码长方面，我们给出了q元最优LRC的最大码长的一个理论上界，并得到一类具有最大码长的最优LRC构造；结构性质方面，我们确定了部分最优LRC的各阶广义Hamming重量，同时确定了一组特定参数下最优LRC的重量分布这一重要结构性质。