For the digital age, data encoding and decoding represents a primary request, without which computer science and information systems would not exist. As a main part of discrete mathematics and computer algebra, the theory of error-correcting codes includes notions, theorems, techniques, and applications of large interest to mathematicians and research engineers. This is the second edition of Introduction to coding theory, a textbook that provides a self-contained introduction to mathematical coding theory and covers, in an accessible format, the results that are used in various areas, such as data communication, cryptography, steganography, statistics, computer science, biology, and so on.

The book is very well structured; its 23 chapters are grouped into three parts. This is a typical book, which has topics from the undergraduate (Part 1) to graduate levels (Parts 2 and 3). The clear, concise, and accessible writing style provides a helpful overview of the theory of error-correcting codes, being especially useful for students. Throughout the book, the theorems are proved and the proofs are detailed. The main goals of the book are to teach students the fundamental concepts in the theory of error-correcting codes and to clearly illustrate them by using examples and exercises for independent practice. This is why it is essential reading for all those involved in the coding theory area, not only students and researchers but also practitioners.

Providing an elementary introduction to coding theory and some typical applications, the first part of this textbook applies these notions to the study of binary and general linear codes, Reed-Solomon codes, recursive constructions of codes, universal hashing, binary Golay code, and 3D codes. Part 1, “An Elementary Introduction to Coding,” is intended for introductory undergraduate courses and includes coding theory fundamentals in its ten chapters: “The Concept of Coding,” “Binary Linear Codes,” “General Linear Codes,” “Singleton Bound and Reed-Solomon Codes,” “Recursive Constructions I,” “Universal Hashing,” “Designs and the Binary Golay Code,” “Shannon Entropy,” “Asymptotic Results,” and “Three-Dimensional Codes, Projective Planes.” The final chapter of Part 1, “Summary and Outlook,” serves to preface Part 2, “Theory and Applications of Codes.”

The second part of the book is devoted to the rigorous presentation of cyclic codes and more recursive constructions, a self-contained introduction to the basics of linear programming, a geometric description of linear codes, and details of error-correcting code applications in statistics and computer science. The mathematical approach permits an exact understanding of these subjects. Part 2 includes seven chapters: “Subfield Codes and Trace Codes,” “Cyclic Codes,” “Recursive Constructions, Covering Radius,” “The Linear Programming Method,” “Orthogonal Arrays in Statistics and Computer Science,” “The Geometric Description of Linear Codes,” and “Additive Codes and Network Codes.”

Part 3, “Codes and Algebraic Curves,” includes “Introduction,” “Function Fields, Their Places and Valuations,” “Determining the Genus,” “AG Codes, Weierstraß Points and Universal Hashing,” and “The Last Chapter.” Compared with the first edition, this part of the book is new and is dedicated to algebraic-geometric (AG) codes. It’s a good choice in the context of this topic; AG codes are fundamental solutions to challenges that arise in communications, cryptography, computer science, and others. A basic construction of AG codes and the properties of some interesting families of examples are subjects of the chapters of Part 3, together with a thorough presentation to some of the basics of the theory of function fields in one variable over a finite field of constants. Based on subjects presented in the book and on new material, the last chapter introduces readers to recent progress in the theory of error-correcting codes and their applications.

Introduction to coding theory provides a gradual and comprehensive exploration of mathematical coding theory and its applications. For those who want to further explore some aspects, the necessary bibliographic references are provided. Being a textbook that provides a self-contained introduction to coding theory, with topics spanning undergraduate to graduate levels, this book represents essential reading for students, researchers, and practitioners.