Channel Coding and Precoding for Linear Network Coding
Michael Cyran · 2017
This thesis considers linear network coding, a capacity achieving method for information dissemination over packet-switched communication networks. Different to conventional solutions to this problem, where distinct data streams are treated separately, the essence of network coding is the mixing of data streams at intermediate nodes of the communication network. In case of linear network coding, intermediate nodes linearly combine their incoming streams. The first part of this thesis (Chapters 2 to 5) focuses on the theoretical foundation of linear network coding. Communication networks are defined, and well-known algorithms for computing the min-cut or for designing the coding coefficients are recapitulated. Finite-field matrix channels, which proved to be a proper end-to-end channel model for communication networks, where intermediate nodes apply linear network coding, are introduced and reviewed. One of the contributions of this thesis is the development of a probabilistic error model for such channels, which covers the error mechanisms within a communication network, the additive white Poisson error rank (AWPER) model. Furthermore, networks with a special arrangement of intermediate nodes, so-called layered networks are discussed. A method called layering, which introduces a layered structure into arbitrary non-layered communication networks is introduced. Based on this equivalent layered structure of networks a forward-backward duality is derived. This duality, which states that the communication from a node A to another node B over a communication network, is dual to the backward communication, i.e., from node B back to node A. It can be seen as the analogon to the famous uplink-downlink duality from classical communications. At the end of the first part, the equivalent layered representation of networks is exploited in order to derive the statistics, and the outage probabilities of network channel matrices in case of random linear network coding. In its second part (Chapters 6 and 7), this theses is concerned with receiver-side and transmitter-side equalization and coding techniques for finite-field matrix channels. First, two well-known receiver-side techniques, lifted Gabidulin codes and channel sounding in combination with error trapping, are analytically and numerically assessed, based on the AWPER error model. Finally, three precoding techniques, which are adopted from the field of precoding for the broadcast channel, are introduced. The first adopted scheme is the linear pre-equalization approach. Furthermore, selection precoding, which is the dual to vector precoding, and differential linear network coding, which can be seen as the dual to differential space-time codes are presented. These schemes are analytically and numerically assessed, and their advantages and disadvantages compared to the well-known schemes are discussed.