Английская Википедия:Introduction to the Theory of Error-Correcting Codes
Шаблон:Short description Шаблон:Italic title Introduction to the Theory of Error-Correcting Codes is a textbook on error-correcting codes, by Vera Pless. It was published in 1982 by John Wiley & Sons,Шаблон:R with a second edition in 1989Шаблон:R and a third in 1998.Шаблон:R The Basic Library List Committee of the Mathematical Association of America has rated the book as essential for inclusion in undergraduate mathematics libraries.Шаблон:R
Topics
This book is mainly centered around algebraic and combinatorial techniques for designing and using error-correcting linear block codes.Шаблон:R It differs from previous works in this area in its reduction of each result to its mathematical foundations, and its clear exposition of the results follow from these foundations.Шаблон:R
The first two of its ten chapters present background and introductory material, including Hamming distance, decoding methods including maximum likelihood and syndromes, sphere packing and the Hamming bound, the Singleton bound, and the Gilbert–Varshamov bound, and the Hamming(7,4) code.Шаблон:R They also include brief discussions of additional material not covered in more detail later, including information theory, convolutional codes, and burst error-correcting codes.Шаблон:R Chapter 3 presents the BCH code over the field <math>GF(2^4)</math>, and Chapter 4 develops the theory of finite fields more generally.Шаблон:R
Chapter 5 studies cyclic codes and Chapter 6 studies a special case of cyclic codes, the quadratic residue codes. Chapter 7 returns to BCH codes.Шаблон:R After these discussions of specific codes, the next chapter concerns enumerator polynomials, including the MacWilliams identities, Pless's own power moment identities, and the Gleason polynomials.Шаблон:R The final two chapters connect this material to the theory of combinatorial designs and the design of experiments,Шаблон:R and include material on the Assmus–Mattson theorem, the Witt design, the binary Golay codes, and the ternary Golay codes.Шаблон:R
The second edition adds material on BCH codes, Reed–Solomon error correction, Reed–Muller codes, decoding Golay codes,Шаблон:R and "a new, simple combinatorial proof of the MacWilliams identities".Шаблон:R As well as correcting some errors and adding more exercises, the third edition includes new material on connections between greedily constructed lexicographic codes and combinatorial game theory, the Griesmer bound, non-linear codes, and the Gray images of <math>\mathbb{Z}^4</math> codes.Шаблон:R
Audience and reception
This book is written as a textbook for advanced undergraduates;Шаблон:R reviewer H. N. calls it "a leisurely introduction to the field which is at the same time mathematically rigorous".Шаблон:R It includes over 250 problems,Шаблон:R and can be read by mathematically-inclined students with only a background in linear algebraШаблон:R (provided in an appendix)Шаблон:R and with no prior knowledge of coding theory.Шаблон:R
Reviewer Ian F. Blake complained that the first edition omitted some topics necessary for engineers, including algebraic decoding, Goppa codes, Reed–Solomon error correction, and performance analysis, making this more appropriate for mathematics courses, but he suggests that it could still be used as the basis of an engineering course by replacing the last two chapters with this material, and overall he calls the book "a delightful little monograph".Шаблон:R Reviewer John Baylis adds that "for clearly exhibiting coding theory as a showpiece of applied modern algebra I haven't seen any to beat this one".Шаблон:R
Related reading
Other books in this area include The Theory of Error-Correcting Codes (1977) by Jessie MacWilliams and Neil Sloane,Шаблон:R and A First Course in Coding Theory (1988) by Raymond Hill.Шаблон:R
References
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