Selected Unsolved Problems in Coding Theory

This original monograph investigates several unsolved problems that currently exist in coding theory. A highly relevant branch of mathematical computer science, the theory of error-correcting codes is concerned with reliably transmitting data over a ‘noisy’ channel. Despite its fairly long history and consistent prominence, the field still contains interesting problems that have resisted solution by some of the most prominent mathematicians of recent decades.

Employing SAGE—a free open-source mathematics software system—to illustrate ideas, this book is intended for graduate students and researchers in algebraic coding theory, especially those who are interested in finding some current unsolved problems. Familiarity with concepts in algebra, number theory, and modular forms is assumed. The work may be used as supplementary reading material in a graduate course on coding theory or for self-study.

• Assmus–Mattson theorem
• Duursma zeta function
• Reed–Muller code
• SAGE
• algebraic coding theory
• algebraic-geometric codes
• block codes
• coding theory
• error-correcting codes
• hyperelliptic polynomial equation
• invariant theory
• modular forms
• quadratic residue
• self-dual codes
• zeta function
• information and communication, circuits

From the reviews:

“This book presents a good number of unsolved problems for those who are interested in the mathematics of the theory of error-correcting codes. It will also be of interest to coding-theorists interested in knowing how to use SAGE to do certain computations with error-correcting codes. … Overall, the book is a treat for coding-theorists in both theoretical and practical aspects.” (Bal Kishan Dass, Mathematical Reviews, Issue 2012 i)

“This is a book for mathematicians interested in learning, in a pleasant and instructive way, some main topics of coding theory. It is also a book for coding theorists interested in learning the program SAGE to perform computations with error-correcting codes. … the book is interesting and enjoyable to read, and offers a good opportunity to learn about some of the most fascinating aspects of the mathematical theory of codes. It is also a good source for learning SAGE in a useful and entertaining way.” (Carlos Munuera, Zentralblatt MATH, Vol. 1239, 2012)

• Chauvenet Hall, Mathematics Department, U.S. Naval Academy, Annapolis, USA

  David Joyner

• Deptartment of Mathematics, University of Louisville, Louisville, USA

  Jon-Lark Kim

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