A Course in Algebraic Error-Correcting Codes

Textbook

Part of the Compact Textbooks in Mathematics book series (CTM)

1. Front Matter
Pages i-xiii
2. Simeon Ball
Pages 1-16
3. Simeon Ball
Pages 17-27
4. Simeon Ball
Pages 29-45
5. Simeon Ball
Pages 47-69
6. Simeon Ball
Pages 71-82
7. Simeon Ball
Pages 83-103
8. Simeon Ball
Pages 105-121
9. Simeon Ball
Pages 123-132
10. Simeon Ball
Pages 133-150
11. Simeon Ball
Pages 151-164
12. Back Matter
Pages 165-177

Introduction

This textbook provides a rigorous mathematical perspective on error-correcting codes, starting with the basics and progressing through to the state-of-the-art. Algebraic, combinatorial, and geometric approaches to coding theory are adopted with the aim of highlighting how coding can have an important real-world impact. Because it carefully balances both theory and applications, this book will be an indispensable resource for readers seeking a timely treatment of error-correcting codes.

Early chapters cover fundamental concepts, introducing Shannon’s theorem, asymptotically good codes and linear codes. The book then goes on to cover other types of codes including chapters on cyclic codes, maximum distance separable codes, LDPC codes, p-adic codes, amongst others. Those undertaking independent study will appreciate the helpful exercises with selected solutions.

A Course in Algebraic Error-Correcting Codes suits an interdisciplinary audience at the Masters level, including students of mathematics, engineering, physics, and computer science. Advanced undergraduates will find this a useful resource as well. An understanding of linear algebra is assumed.

Keywords

Coding theory Coding theory error correction Error-correcting codes Algebraic error-correcting codes Shannon's theorem Shannon-Hartley theorem Finite fields Block codes Linear codes Linear code coding theory Cyclic code Cyclic code error detection MDS codes Algebraic geometric codes p-adic codes LDPC codes Expanders Reed-Muller codes Reed-Muller error-correcting codes Masters level error-correcting codes

Authors and affiliations

1. 1.Department of MathematicsPolytechnic University of CataloniaBarcelonaSpain

Simeon Ball is Senior Lecturer of Mathematics at Universitat Politècnica de Catalunya in Barcelona, Spain. He has been invited speaker at many international conferences, as well as serving on the scientific and organising committee for the Fq series of conferences. His research interests include classical and quantum error-correcting codes, incidence problems in real and finite geometries, graphs and semifields, and is particularly focused on applying geometrical and algebraic methods to these combinatorial objects. He also serves on the editorial board of the Journal of Geometry, having previously served on the editorial board of Designs, Codes and Cryptography and Finite Fields and Their Applications.

Bibliographic information

• Book Title A Course in Algebraic Error-Correcting Codes
• Authors Simeon Ball
• Series Title Compact Textbooks in Mathematics
• Series Abbreviated Title Compact Textbooks in Mathematics
• DOI https://doi.org/10.1007/978-3-030-41153-4
• Copyright Information Springer Nature Switzerland AG 2020
• Publisher Name Birkhäuser, Cham
• eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
• Softcover ISBN 978-3-030-41152-7
• eBook ISBN 978-3-030-41153-4
• Series ISSN 2296-4568
• Series E-ISSN 2296-455X
• Edition Number 1
• Number of Pages XIII, 177
• Number of Illustrations 7 b/w illustrations, 5 illustrations in colour
• Topics
• Buy this book on publisher's site

Reviews

“The style is clear and the topics are well presented, which makes the understanding of the subject approachable even for students coming from applied sciences. This is a remarkable textbook for a self-contained introduction to the theory of error-correcting codes and some of their modern topics.” (Matteo Bonini, zbMATH 1454.94142, 2021)