本书的主要内容包括编码学的数学基础，编码的分类及原理，网络编码的概念和原理，网络编码在网络容量域、网络**和分布式存储这3个重要领域的应用。本书把理论与实践相结合，既清晰地阐述了网络编码的基本概念和原理，又详尽地解释了网络编码的算法实现和系统设计；既对网络编码和路由技术进行了特点分析和优缺点比较，又对网络编码在提高网络容量，网络**和网络存储等方面的效能进行了深入解析。本书有助于理论研究者理解网络编码的内涵，也有助于工程实践者了解下一代网络的发展趋势。

Chapter 1 Coding Mathematics 1 1.1 Set Theory 1 1.1.1 The Rudiments of Set Theory 2 1.1.2 The Operation of Sets 4 1.1.3 Mapping 6 1.1.4 The Equivalence of Sets 7 1.1.5 The Classification of Sets 9 1.2 Abstract Algebra 11 1.2.1 Group 11 1.2.2 Ring 14 1.2.3 Field 19 1.2.4 Finite Field 21 1.2.5 Extension Fields 23 1.3 Vector Space 28 1.4 Probability Theory 30 1.4.1 Random Events and Probability 30 1.4.2 Random Variables 33 1.4.3 Numerical Characteristics of Random Variables 40 Chapter 2 Classical Information Theory 43 2.1 Introduction 43 2.2 Discrete Information Source and Discrete Entropy 46 2.2.1 The Discrete Entropy 47 2.2.2 The Average Mutual Information 51 2.2.3 Discrete Lossless Source Coding 52 2.2.4 Huffman Code 55 2.3 Discrete Channel and Channel Capacity 56 2.3.1 The Classification of Channel 56 2.3.2 Channel Capacity 59 2.4 Noisy Channel Coding 64 2.4.1 Decoding Rule of Channels 64 2.4.2 Channel Coding 70 2.5 Linear Block Code 73 2.5.1 G Matrix and H Matrix 76 2.5.2 Error Detection and Correction 78 2.5.3 Standard Array and Syndrome Decoding 82 Chapter 3 Network Coding 84 3.1 Fundamentals of Network Coding 84 3.2 Linear Network Coding 91 3.2.1 Deterministic LNC 91 3.2.2 Random LNC 94 3.3 Polynomial Time Algorithm for Linear Network Coding 96 3.3.1 Polynomial Time Algorithm 96 3.3.2 LNCBSD 98 Chapter 4 Network Capacity 109 4.1 Network Flows 109 4.2 Network Rate Region and Uniform Fractional Network Model 111 4.2.1 Network Rate Region 111 4.2.2 Uniform Fractional Network Model 115 4.3 Routing Rate Region 118 4.3.1 Multiple Unicast Routing Networks 119 4.3.2 Single Multicast Routing Networks 125 4.3.3 Mixed Patterns of Unicast and Multicast 139 4.4 Coding Rate Region 144 4.4.1 Theoretical Solution of the Coding Rate Region 144 4.4.2 The Cut Set Bound of Coding Rate Region 147 Chapter 5 Network Security Coding 152 5.1 Basics of Network Security 152 5.2 Security of Network Coding ?C An Overview 156 5.3 NC Security against Passive Attacks 160 5.3.1 Wiretap Model and Security Grade 160 5.3.2 Basic Security 163 5.3.3 Weak Security 167 5.3.4 Perfect Security 174 5.3.5 Strong Security 183 5.3.6 Security of Multisource Networks 185 5.3.7 Security Against TAAs 188 5.4 Secure Error Correcting Codes 195 5.5 NC Security against Active Attacks 198 5.6 Future Developing of NC Security 201 Chapter 6 Network Storage Coding 203 6.1 Basics of Network Storage Code 203 6.2 A Network Coding Based Cloud Storage Scheme 209 6.3 Subspace Storage Code 215 6.3.1 The Theoretical Framework of Subspace Storage Code 217 6.3.2 A Subspace Regenerating Storage Code with Minimum Decoding Bandwidth 219 6.3.3 Comparisons and Discussions 229 References 231