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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、网络安全和分布式存储这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，网络安全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 – 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

1. Contents and Objectives
Chapter 1 introduced the knowledge of coding mathematics, including set theory, abstract algebra (group, ring, field, and finite field), vector space and probability theory (random events, probability, random variables, expectation, and variance).
Chapter 2 introduced the classical information theory, including the concept of information, entropy, mutual information, etc. Moreover, the basic ideas of source coding and channel coding are expounded. Huffman code and linear block code, which are two typical representatives for source coding and channel coding, are explained in detail.
Chapter 3 introduced the fundamentals of network coding, including the concept of linear network coding, and the comparison between network coding and routing strategy. The classification of deterministic network coding and random network coding is given and interpreted. Still, the operation of polynomial time algorithm and linear network coding based on subtree decomposition are expounded.
Chapter 4 introduced the concept of network rate region, which plays the similar role of channel capacity. However, the mathematics model, analysis method, and complexity of them are significantly different. This chapter gives a couple of analysis methods for calculating routing rate region and coding rate region.
Chapter 5 introduced the basics of network security, including security attacks, services, and approaches. The situation of network coding in front of passive attacks and active attacks is analyzed and compared. Specifically, the network coding against passive attacks is classified into four grades: basic security, weak security, perfect security, and strong security. Many schemes for each security grade are given and analyzed. The secure coding schemes against traffic analysis attacks and pollution attacks are illustrated. Moreover, the basic idea and implementation of error correction code are also explained in this chapter.
Chapter 6 introduced the basics of network storage. Specially, the concepts of regenerating code, repair bandwidth, decoding bandwidth, and update bandwidth are elucidated. Moreover, a scheme of (n, k, m) code and a type of subspace storage code with minimum decoding bandwidth are proposed and expounded in detail. 
Chapter 1 to Chapter 3 of this book were completed by Liu Yantao (Jiaying University). Chapter 4 to Chapter 5 was completed by Qin Na (Jiaying University). Chapter 6 was completed by Wang Xuebing (Tianjin University, Shandong Institute of Petroleum and Chemical Technology). The entire book was compiled by Liu Yantao.
One objective of this book is to show and explain the classic Shannon information theory for readers, help readers overcome the difficulties of abstract concepts, obscure theorems, tedious derivation, and explain the meaning behind the theorems and definitions of information theory, illustrate where and how to apply this theory so as to connect information theory with engineering applications.
The other objective of this book is to open a door and guide readers into the relatively new research field of network information theory. Network information theory studies the laws of information transmission, processing, storage, control, and utilization in the network. Modern network information theory has been developed for only 20 years since its inception in millennium, its theoretical system is far less mature than the classic Shannon information theory, so this book is intended to serve as an introduction rather than a survey. It introduces the basic concepts of network coding and emphasizes its application on network rate region, network security and network storage. Whereas, many other topics, such as nonlinear network coding, network tomography and topological inference, network convolutional code, are not included.
2. Audience
This monograph could be used by undergraduate students, graduate students and teachers majored in electronic information engineering, communication engineering, computer science, and network engineering. It is also applicable for the researchers and engineers in the relevant fields such as information theory, information and communication technology, networking, etc. 
3. Acknowledgements
First, in the writing process of this monograph, the author read a large number of journal papers and conference papers from databases, including IEEE, ACM, IET, Elsevier, Springer, IEICE, etc. These papers not only broadened the author’s horizons, promoted the author’s scientific research work, but also greatly contributed to the completion of this monograph. These citation sources are listed in the references at the end of the book. We would like to express our sincerest respect and gratitude to the authors of the references.
Next, the authors would like to show great appreciation to the colleagues from Jiaying University, including Wang Xiaozeng, Huang Jiexian, Zhong Qinghu, Zhu Xiangqing, Yan Lei, Hou Yueen, Guo Jianghong, Zhao Xin, Zhang Haifeng, Zhong Baohua, et al.
Finally, this monograph was supported in part by the following funds.
(1) 2022 Guangdong Basic and Applied Basic Research Foundation (No.2022A1515011345).
(2) 2021 Science Platforms and Projects for Universities of Guangdong Province (No. 2021KTSCX124). 
(3) 2022 Meizhou City Social Development Science and Technology Planning Project (No. 2022B110).
(4) 2023 Scientific Research Project of Jiaying University (No.2023KJY020).
(5) 2021 Higher Education and Teaching Innovation Project of Guangdong Province.
(6) 2020 Course Ideological and Political Education Project of Guangdong Province.
(7) 2021 Course Ideological and Political Education Project of Guangdong Institute of Higher Education (No.X-KCSZ2021178).
(8) 2022 Funds of the Section of Laboratory Management of Guangdong Institute of Higher Education (No.GDJ20220290).
(9) 2021 Teaching Quality and Innovation Engineering Project of Jiaying University.
(10) 2020 Higher Education and Teaching Innovation Project of Jiaying University (No. JYJG20200102).
(11) 2022 Higher Education and Teaching Innovation Project of Jiaying University (No. JYJG2022222).
(12) 2022 Course Ideological and Political Education Project of Jiaying University.
(13) 2023 Guangdong Province Education Science Planning Program (Special Project for Higher Education)(No.2023GXJK486).
(14) 2023 First Class Undergraduate Course of Jiaying University.
(15) 2023 Engineering Center of Meizhou City(No.2023GC03).

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网络编码——容量、安全与存储（英文版）