Information and Coding Theory

Information and coding theory are increasingly popular topics in undergraduate curricula, not only in mathematics but also in computer science and engineering. Written by the authors of Elementary Number Theory, the best selling SUMS title, this book introduces the basic ideas at undergraduate level and is the first book to consider both topics, in depth, in one volume.

Written by the authors of "Elementary Number Theory", a hugely popular SUMS title Information and Coding Theory are increasingly popular topics in undergraduate curricula: this is the first book to cover both topics in-depth in one volume Includes supplementary material: sn.pub/extras

Contenu

1. Source Coding.- 1.1 Definitions and Examples.- 1.2 Uniquely Decodable Codes.- 1.3 Instantaneous Codes.- 1.4 Constructing Instantaneous Codes.- 1.5 Kraft's Inequality.- 1.6 McMillan's Inequality.- 1.7 Comments on Kraft's and McMillan's Inequalities.- 1.8 Supplementary Exercises.- 2. Optimal Codes.- 2.1 Optimality.- 2.2 Binary Huffman Codes.- 2.3 Average Word-length of Huffman Codes.- 2.4 Optimality of Binary Huffman Codes.- 2.5 r-ary Huffman Codes.- 2.6 Extensions of Sources.- 2.7 Supplementary Exercises.- 3. Entropy.- 3.1 Information and Entropy.- 3.2 Properties of the Entropy Function.- 3.3 Entropy and Average Word-length.- 3.4 Shannon-Fano Coding.- 3.5 Entropy of Extensions and Products.- 3.6 Shannon's First Theorem.- 3.7 An Example of Shannon's First Theorem.- 3.8 Supplementary Exercises.- 4. Information Channels.- 4.1 Notation and Definitions.- 4.2 The Binary Symmetric Channel.- 4.3 System Entropies.- 4.4 System Entropies for the Binary Symmetric Channel.- 4.5 Extension of Shannon's First Theorem to Information Channels.- 4.6 Mutual Information.- 4.7 Mutual Information for the Binary Symmetric Channel.- 4.8 Channel Capacity.- 4.9 Supplementary Exercises.- 5. Using an Unreliable Channel.- 5.1 Decision Rules.- 5.2 An Example of Improved Reliability.- 5.3 Hamming Distance.- 5.4 Statement and Outline Proof of Shannon's Theorem.- 5.5 The Converse of Shannon's Theorem.- 5.6 Comments on Shannon's Theorem.- 5.7 Supplementary Exercises.- 6. Error-correcting Codes.- 6.1 Introductory Concepts.- 6.2 Examples of Codes.- 6.3 Minimum Distance.- 6.4 Hamming's Sphere-packing Bound.- 6.5 The Gilbert-Varshamov Bound.- 6.6 Hadamard Matrices and Codes.- 6.7 Supplementary Exercises.- 7. Linear Codes.- 7.1 Matrix Description of Linear Codes.- 7.2 Equivalence ofLinear Codes.- 7.3 Minimum Distance of Linear Codes.- 7.4 The Hamming Codes.- 7.5 The Golay Codes.- 7.6 The Standard Array.- 7.7 Syndrome Decoding.- 7.8 Supplementary Exercises.- Suggestions for Further Reading.- Appendix A. Proof of the Sardinas-Patterson Theorem.- Appendix B. The Law of Large Numbers.- Appendix C. Proof of Shannon's Fundamental Theorem.- Solutions to Exercises.- Index of Symbols and Abbreviations.