On the relation between linear dispersion and generic network code.

Kwok Pui Wing.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 66-67).Abstracts in English and Chinese.Abstract --- p.iAbstract (Chinese Version) --- p.iiAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 2 --- Linear Network Coding --- p.7Chapter 2.1 --- Single Source Network Coding --- p.8Chapter 2.2 --- Descriptions of Linear Network Codes --- p.9Chapter 2.3 --- Desirable Properties of Linear Network Codes --- p.12Chapter 2.4 --- Linear Network Codes Constructions --- p.14Chapter 3 --- Node-based Characterization --- p.16Chapter 3.1 --- Channel-based characterization --- p.16Chapter 3.2 --- A Necessary Condition for the Existence of Linear Network Codes --- p.17Chapter 3.3 --- Insufficiency of the condition --- p.22Chapter 4 --- Relation between Linear Network Codes --- p.25Chapter 4.1 --- Relation between Multicast and Broadcast --- p.26Chapter 4.1.1 --- Auxiliary Graph --- p.26Chapter 4.2 --- Relation between Broadcast and Dispersion --- p.29Chapter 4.2.1 --- Expanded Graph --- p.29Chapter 4.3 --- Relation between Dispersion and Generic Net- work Code --- p.31Chapter 4.3.1 --- Edge Disjoint Path --- p.31Chapter 4.3.2 --- Path Rearrangement --- p.34Chapter 4.3.3 --- Extended Graph --- p.50Chapter 5 --- Upper Bound on the Size of the Base Field --- p.57Chapter 5.1 --- Base Field Size Requirement --- p.58Chapter 5.1.1 --- Linear Multicast --- p.58Chapter 5.1.2 --- Linear Broadcast --- p.58Chapter 5.1.3 --- Linear Dispersion --- p.59Chapter 5.1.4 --- Generic Network Code --- p.60Chapter 5.2 --- Upper Bounds Comparison for Generic Network Code --- p.61Chapter 6 --- Future Work --- p.62Bibliography --- p.6

Scalar-linear Solvability of Matroidal Networks Associated with Representable Matroids

We study matroidal networks introduced by Dougherty et al. We prove the converse of the following theorem: If a network is scalar-linearly solvable over some finite field, then the network is a matroidal network associated with a representable matroid over a finite field. It follows that a network is scalar-linearly solvable if and only if the network is a matroidal network associated with a representable matroid over a finite field. We note that this result combined with the construction method due to Dougherty et al. gives a method for generating scalar-linearly solvable networks. Using the converse implicitly, we demonstrate scalar-linear solvability of two classes of matroidal networks: networks constructed from uniform matroids and those constructed from graphic matroids.Comment: 5 pages, submitted to IEEE ISIT 201

Network Coding for Multi-Resolution Multicast

Multi-resolution codes enable multicast at different rates to different receivers, a setup that is often desirable for graphics or video streaming. We propose a simple, distributed, two-stage message passing algorithm to generate network codes for single-source multicast of multi-resolution codes. The goal of this "pushback algorithm" is to maximize the total rate achieved by all receivers, while guaranteeing decodability of the base layer at each receiver. By conducting pushback and code generation stages, this algorithm takes advantage of inter-layer as well as intra-layer coding. Numerical simulations show that in terms of total rate achieved, the pushback algorithm outperforms routing and intra-layer coding schemes, even with codeword sizes as small as 10 bits. In addition, the performance gap widens as the number of receivers and the number of nodes in the network increases. We also observe that naiive inter-layer coding schemes may perform worse than intra-layer schemes under certain network conditions.Comment: 9 pages, 16 figures, submitted to IEEE INFOCOM 201