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Codes, Cryptology and Curves with Computer Algebra

Ruud Pellikaan, Xin-Wen Wu, Stanislav Bulygin and Relinde Jurrius
Cambridge University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Darren Glass
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As the story goes, Richard Hamming was working on state of the art computers in the 1940s and getting frustrated that the machines would always freeze up when there was an error on one of his punch cards, and this is what caused him to envision the idea of computers being able to detect, and even correct, errors in their codes. Over the next three-quarters of a century, communications have become increasingly digital and the mathematical theory of error-correcting codes has exploded far beyond what Hamming ever envisioned. It is now a field that incorporates linear algebra, group theory, Galois theory, combinatorics, and algebraic geometry — and that is before one gets to the engineering and algorithmic questions!

The book under review is intended as an introduction to the field for beginning graduate students. The authors do a good job of covering a wide range of topics and keeping the discussion detailed while still as elementary as one can hope to make it. They cover various constructions of error-correcting codes, as well as different invariants that one might use to classify them. The four authors trade off writing and co-writing different chapters, which leads to some strange overlaps and gaps if one reads it as a cohesive textbook, but which also allows a reader to get different perspectives on the same topics.

It is worth saying that the title of the book is somewhat misleading, as “cryptography” makes up a quarter of the title but only features in one chapter of the book, and most of that is really about some of the (admittedly exciting) ways that error-correcting codes are now being used to develop new cryptosystems. Similarly, there is only one chapter that discusses computer algebra at length, although that chapter gives a very nice introduction to ways one could use programs such as Singular, Magma, Gap, and Sage in order to implement the ideas in other parts of the book, many of which are discussed as abstract algorithms earlier.

Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose interests range from cryptography to Galois theory to graph theory. He can be reached at

Preface Ruud Pellikaan
1. Error-correcting codes Ruud Pellikaan and Xin-Wen Wu
2. Code constructions and bounds on codes Ruud Pellikaan and Xin-Wen Wu
3. Weight enumeration Relinde Jurrius, Ruud Pellikaan and Xin-Wen Wu
4. Cyclic codes Ruud Pellikaan
5. Polynomial codes Ruud Pellikaan
6. Algebraic decoding Ruud Pellikaan and Xin-Wen Wu
7. Complexity and decoding Stanislav Bulygin, Ruud Pellikaan and Xin-Wen Wu
8. Codes and related structures Relinde Jurrius and Ruud Pellikaan
9. Cryptology Stanislav Bulygin
10. Gröbner bases for coding and cryptology Stanislav Bulygin
11. Codes on curves Ruud Pellikaan
12. Coding and cryptology with computer algebra Stanislav Bulygin