基于相对广义Hamming重量研究分布式存储中的数据恢复和数据安全

In recent years, “Big Data” has become a new ubiquitous term. Big Data is transforming “science, engineering, medicine, healthcare, finance, business, and the society itself”. Distributed storage is obviously an important part of it. However, in distributed storage systems, one problem is that the reliability is often maintained using erasure coding, replication and regenerating code, which cannot be designed to meet multiple requirements such as partial recovery, trade-off between security and recovery, practical list decoding, and material performance. Recently, network coding techniques have been suggested in addressing these challenges, establishing that maintenance bandwidth can be reduced by orders of magnitude compared to standard erasure codes. In this proposal, we prefer Relative Generalized Hamming Weight to deal with the problem, which also works on network coding.. The dimension/length profile (DLP) and the generalized Hamming weight (GHW) hierarchy of a linear code play an important role in coding theory, especially in trellis complexity, secure communication, multiple access communication and puncturing codes. It is worth noting that trellis complexity has been applied to the satellite system of NASA. By investigating the wiretap channel of type II with illegitimate parties, Luo et.al introduced the relative generalized Hamming weight (RGHW) hierarchy of a linear code and a subcode, which is a generalization of GHW hierarchy. Some equivalent concepts are relative dimension/length profile (RDLP), inverse relative dimension/ length profile (IRDLP) and relative length/dimension profile (RLDP). Recently, Zhang et al. extended RGHW hierarchy to secure network coding and described the equivocation to an eavesdropper in the wiretap network II with side information leakage.. Here we study important properties (maybe asymptotical) of RGHW, its equivalent concepts, its coding constructions, and then apply them to the (full or partial) recovery ability of linear coding data storage system combining with data security. For achieving the multiple requirements, we focus on not only the coding rate, the data security, but also on various storage materials, practical list decoding and their realization.

近年来，分布式存储技术在国内外掀起了研究热潮，也成为国家发展规划中大数据研究的重要一环。然而目前分布式存储系统中，一般很少考虑客户的多样化需求，例如结点部份（删截）数据的快速恢复问题、数据恢复同数据安全的平衡问题、满足多解需求的编码构造问题、存储介质性质问题等等，也较难精确刻画满足上述需求的模型结构。这正是此次课题申请需要解决的关键问题。. 申请人将利用与合作者提出的相对广义Hamming 重量概念，对基于直接线性编码或网络线性编码的分布式存储编码模式，提出满足上述理论与应用多样化需求、适应性强的存储方案与理论依据。另外，相对广义Hamming 重量本身的研究还可以对通信代数编码一系列经典性质进行推广，从子码的角度更好地理解原码的构造。

近年来，由再生码技术和局部修复码技术而引领的一批网络存储修复技术成为工业界和学术界的研究热点。但问题同样很突出：网络异构的普遍性弱化了再生码理论的应用基础，码长设计的局限性困住了局部修复码的适用范围。针对这些多样化的应用问题，项目组通过一系列科研工作，为集群和散点异构网络提出了再生码修复带宽和存储容量平衡理论；为局部修复码摆脱符号集合的束缚提出了新的码长构造理论；同时，项目组从广义Hamming重量出发，揭示了单纯最小带宽修复的解析机理。围绕这三个主要研究内容，项目组还关注了安全约束下的存储修复技术，以及存储介质的纠错特性。. 对于异构网络再生码修复技术，我们研究了多散点集群分布式存储系统模型，该模型又可称作异构集群分布式存储系统模型。我们给出了多散点情况下的系统容量以及存储修复带宽折中界。此外，我们同样给出了多散点模型下的MSR 码构造实例。 . 对于局部修复码技术，其最优修复性能往往依赖于Singleton类型的界，构造大长度的局部修复码是业内公认的学术挑战。项目组给出了一类不受限于符号集大小的循环码码长设计方案，并在具有特殊生成和校验多项式的循环码上得到了实现。. 此外，当仅仅考虑最小带宽存储修复技术时，项目组发现利用广义Hamming重量可以严格解析其修复能力，从而为贯通各类修复技术提供一个统一的数学工具，特别是能够在安全约束下继续清晰刻画其修复能力。深刻理解两个研究热点及一个公共基础的理论体系，有助于申请人在上海市大数据试验场和SKA大数据处理领域拓展良好的存储修复应用前景。

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