Algebraic Coding Theory

This is the revised edition of Berlekamp’s famous book, “Algebraic Coding Theory”, originally published in 1968, wherein he introduced several algorithms which have subsequently dominated engineering practice in this field. One of these is an algorithm for decoding Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that subsequently became known as the Berlekamp–Massey Algorithm. Another is the Berlekamp algorithm for factoring polynomials over finite fields, whose later extensions and embellishments became widely used in symbolic manipulation systems. Other novel algorithms improved the basic methods for doing various arithmetic operations in finite fields of characteristic two. Other major research contributions in this book included a new class of Lee metric codes, and precise asymptotic results on the number of information symbols in long binary BCH codes.

Selected chapters of the book became a standard graduate textbook.

Both practicing engineers and scholars will find this book to be of great value.

Readership: Researchers in coding theory and cryptography, algebra and number theory, and software engineering.

Table of Contents

Chapter 1. Basic Binary Codes
Chapter 2. Arithmetic Operations Modulo an Irreducible Binary Polynomial
Chapter 3. The Number of Irreducible q-ary Polynomials of Given Degree
Chapter 4. The Structure of Finite Fields
Chapter 5. Cyclic Binary Codes
Chapter 6. The Factorization of Polynomials Over Finite Fields
Chapter 7. Binary BCH Codes for Correcting Multiple Errors
Chapter 8. Nonbinary Coding
Chapter 9. Negacyclic Codes for the Lee Metric
Chapter 10. Gorenstein-Zierler Generalized Nonbinary BCH Codes for the Hamming Metric
Chapter 11. Linearized Polynomials and Affine Polynomials
Chapter 12. The Enumeration of Information Symbols in BCH Codes
Chapter 13. The Information Rate of the Optimum Codes
Chapter 14. Codes Derived by Modifying or Combining Other Codes
Chapter 15. Other Important Coding and Decoding Methods
Chapter 16. Weight Enumerators