15 has been polished and simplified.
Chapter 16 includes background and full details of the Berlekamp–Massey algorithm.
Chapter 17 has details of the WKdm algorithms.
Chapter 18 outlines the proof by one of the authors on a long‐standing conjecture regarding the next‐to‐minimum weights of Reed–Muller codes.
Chapter 21 features a fresh approach to cyclic linear codes and culminates with a new user‐friendly proof of a powerful result on the periodicity of shift registers in Peterson and Weldon, [PW72].
The study of MDS codes leads to a very interesting and basic “inverse” problem in linear algebra over any field. It could be discussed in a first year linear algebra class. See Chapter 23 for the details.
Chapter 24 introduces a new hash function and improvements to the main algorithm in the chapter.
Chapter 25 brings readers of this book to the very forefront of research by exhibiting infinitely many new identities for the Shannon function.
Chapter 26 features a simple new proof of the security of Bitcoin in the matter of double spending, avoiding the assumptions of the approximation by a continuous random variable in the original paper by Nakamoto ([Nak08]).
Chapter 27 discusses privacy and security concerns relating to the Internet of Things (IoT). Important questions include: Who has access to the information that your smart device is collecting? Could someone remotely access your smart device?
Chapter 28 focuses on the availability of data stored in the cloud and on homomorphic encryption, which allows computations to be done on data while it is in an encrypted form.
Chapter 29 features another approach to MDS codes and, we hope, a very interesting discussion of the venerable topic of mutually orthogonal latin squares. There are also exercises in modular arithmetic, finite fields, linear algebra, and other topics to elucidate theoretical results in previous chapters, along with solutions.
Hardcover and eBook
The second edition will be available both as a hardcover book and as an eBook. The content will be the same in both. Besides traditional formatting for items in the bibliography, most of the items have accompanying URLs.
The eBook will have clickable links, including links to chapter and section numbers, to theorem numbers, from problems to their solutions, and to items in the bibliography. The URLs in the bibliography will also be clickable in the eBook.
Numbering of Definitions, Examples, Results.
When referring to a definition or result, we list the chapter number, a dot and then a number from an increasing counter for that chapter. For instance, Example 10.7 is the seventh numbered item in Chapter 10. Theorem 10.8 comes after Example 10.7 and is the eight such numbered item in Chapter 10.
Numbering of Problems, Solutions.
Most chapters have a section called Problems followed immediately by a corresponding section called Solutions at the end of the chapter. Problems and Solutions at the end of the chapter have their own counters. So, Problem 10.6 is the sixth problem in the Problems section (Section 10.15) of Chapter 10 and Solution 10.6 has the solution to that problem. It can be found in the subsequent section (Section 10.16).
Numbering of Equations.
Equation numbers follow their own counter for each chapter. For example, Equation (9.7) is the seventh equation in Chapter 9.
Acknowledgments for the Second Edition
The third author is extremely grateful to the first two authors for inviting him to be a co‐author on the second edition! Thank you so much!
We are extremely grateful to a few individuals for their help with the second edition. We thank Professor Dan McQuillan from the Department of Mathematics at Norwich University in Vermont for a careful reading and many improvements to many chapters in the second edition. We thank Joy McQuillan for a careful reading and improvements to several chapters. We are indebted to Professor Sumesh Philip from the School of Information Technology at Illinois State University in Illinois for many significant improvements to the new content in the second edition. We thank Professor David Wehlau of the Department of Mathematics and Computer Science at the Royal Military College of Canada and the Department of Mathematics and Statistics at Queen's University in Kingston, Canada for valuable comments. We also thank Dr. Valery Ipatov from Petersburg State Electrotechnical University in Russia for numerous corrections to the first edition, and Burt Wilsker for corrections to the first edition. These were incorporated into the second edition.
We thank the Wiley staff including Kimberly Monroe‐Hill, Kathleen Pagliaro, Blesy Regulas, Linda Christina E, Mindy Okura‐Marszycki, and Kathleen Santoloci for their help with the second edition. We also thank Wiley staff Gayathree Sekar, Becky Cowan, and Aileen Storry.
Book Website
The website for the book is
It will be a repository for additional information and updates.
We have done our best to correct the errors but, inevitably, some will remain. We invite our readers to submit errors to [email protected] . We will post them, with attribution, on the website along with other clarifications as they arise.
About the Authors
Aiden A. Bruen was born in Galway, Ireland. He read mathematics for his Undergraduate and Master's degree in Dublin and received his Doctorate at the University of Toronto, supervised by F.A. Sherk. At Toronto, he also worked with H.S.M. Coxeter, E. Ellers, and A. Lehman. Dr. Bruen is an Adjunct Research Professor at Carleton University and a Professor Emeritus at the University of Western Ontario.
Mario A. Forcinito was born in Buenos Aires, Argentina where he took his Bachelor's degree in Engineering. He obtained his doctorate in Engineering at
