拟阵在网络编码中的应用

发布时间：2018-05-26 04:41

本文选题：网络编码 + 网络纠错码　；参考：《西安电子科技大学》2014年博士论文

【摘要】：网络编码理论的创立是21世纪初通信和信息论领域的重大突破，其核心思想是允许网络中间节点对输入信息做线性或非线性的编码处理后再转发。网络编码是一种融合了路由和编码的信息传输机制，已经被证明在提高吞吐量、数据安全、鲁棒性、普适性、负载均衡及低计算复杂性等方面具有很大优势。目前，网络编码已成为网络和通信领域的研究热点。在此带动之下，许多数学方法被应用到网络编码理论的研究当中，主要包括代数、图论、拟阵论、组合与优化等等。网络编码的一个基本问题是理解容量区域并研究达到容量界的编码类型。利用拟阵这一数学工具构造有效的网络编码方案及研究多源多宿网络编码的容量区域成为当前网络编码领域重要的研究课题之一。本文着重研究了拟阵在确定编码容量区域的边界、构造线性网络纠错编码等方面的应用，取得的主要成果包括以下几个方面： 1.根据Dougherty等人提出的构造可拟阵化网络的方法和步骤得到了与向量拟阵R8相关的网络，，并利用Ingleton不等式和张-扬非香农型信息不等式得到了该网络编码容量的一个上界。 2.利用扩展网络与扩展全局编码核等概念，刻画了线性网络纠错码与可表示拟阵的关系。根据线性网络纠错码的本质特征，改进了Prasad等提出的可拟阵化纠错网络的定义，将其推广到线性网络纠错码在不同的信宿节点（集）具有不同的纠错能力的情形。随后，并研究了单信源可拟阵化网络与线性多播/线性广播/线性扩散网络纠错MDS码的关系。 3.提出了一种基于二阶射影线性群PGL(2, p)的子群H上的双层群网络编码方法，证明了群直积H n中的双层群网络码可利用加法群Z p与Zp1上的n-长线性码构造而成。最后，用一个可拟阵化网络的实例说明了所提出的编码方案在可达容量区域方面的优势。
[Abstract]:The establishment of network coding theory is a great breakthrough in the field of communication and information theory at the beginning of the 21st century. Its core idea is to allow network intermediate nodes to do linear or nonlinear coding processing of input information and then forward it. Network coding is a kind of information transmission mechanism which combines routing and coding. It has been proved to have great advantages in improving throughput, data security, robustness, universality, load balancing and low computational complexity. At present, network coding has become a research hotspot in the field of network and communication. As a result, many mathematical methods have been applied to the research of network coding theory, including algebra, graph theory, matroid theory, combination and optimization, and so on. One of the basic problems of network coding is to understand the capacity region and study the coding types that reach the capacity bound. Using matroid as a mathematical tool to construct an effective network coding scheme and to study the capacity region of multi-source and multi-home network coding has become one of the most important research topics in the field of network coding. This paper focuses on the application of matroids in determining the boundary of coding capacity region and constructing linear network error correction coding. The main results obtained include the following aspects: 1. According to the method and steps of constructing matroid network proposed by Dougherty et al, the network related to vector matroid R8 is obtained, and an upper bound of the coding capacity of the network is obtained by using Ingleton inequality and Zhang Yangfei Shannon type information inequality. 2. By using the concepts of extended network and extended global coding kernel, the relationship between linear network error correction codes and representable matroids is described. According to the essential characteristics of error-correcting codes in linear networks, the definition of matroid error-correcting networks proposed by Prasad et al is improved, which is extended to the cases where linear network error-correcting codes have different error-correcting capabilities at different lock-in nodes (sets). Then, the relationship between single source matroid network and linear multicast / linear broadcast / linear diffusion network error correction MDS codes is studied. 3. A bilevel group network coding method on subgroup H based on second-order projective linear group PGL2, p) is proposed. It is proved that bilayer group network codes in group direct product H _ n can be constructed by using additive group Z _ p and Zp1 n-long linear codes. Finally, an example of matroid network is given to illustrate the advantages of the proposed coding scheme in the area of reachability.
【学位授予单位】：西安电子科技大学
【学位级别】：博士
【学位授予年份】：2014
【分类号】：TN911.2;O157.5

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