## Introduction to Coding TheoryIt is gratifying that this textbook is still sufficiently popular to warrant a third edition. I have used the opportunity to improve and enlarge the book. When the second edition was prepared, only two pages on algebraic geometry codes were added. These have now been removed and replaced by a relatively long chapter on this subject. Although it is still only an introduction, the chapter requires more mathematical background of the reader than the remainder of this book. One of the very interesting recent developments concerns binary codes defined by using codes over the alphabet 7l.4• There is so much interest in this area that a chapter on the essentials was added. Knowledge of this chapter will allow the reader to study recent literature on 7l. -codes. 4 Furthermore, some material has been added that appeared in my Springer Lec ture Notes 201, but was not included in earlier editions of this book, e. g. Generalized Reed-Solomon Codes and Generalized Reed-Muller Codes. In Chapter 2, a section on "Coding Gain" ( the engineer's justification for using error-correcting codes) was added. For the author, preparing this third edition was a most welcome return to mathematics after seven years of administration. For valuable discussions on the new material, I thank C.P.l.M.Baggen, I. M.Duursma, H.D.L.Hollmann, H. C. A. van Tilborg, and R. M. Wilson. A special word of thanks to R. A. Pellikaan for his assistance with Chapter 10. |

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### Contents

CHAPTER | 1 |

CHAPTER 2 | 20 |

CHAPTER 3 | 33 |

CHAPTER 12 | 34 |

CHAPTER 4 | 47 |

CHAPTER 5 | 64 |

CHAPTER 6 | 81 |

CHAPTER 7 | 111 |

CHAPTER 8 | 127 |

CHAPTER 10 | 148 |

CHAPTER 11 | 167 |

Arithmetic Codes | 173 |

Hints and Solutions to Problems | 195 |

218 | |

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### Common terms and phrases

affine AG(m algebraic geometry BCH code binary code binary image Chapter check polynomial code of length code over F codewords coding theory coefficients columns consider construction convolutional codes coordinates Corollary corresponding cosets cyclic code decoding defined Definition denote designed distance differential dimension divisor equation equivalent example extended Hamming code factor following theorem Goppa codes GRM code Hamming code Hence idempotent implies integer irreducible polynomials Lemma Let q linear code linear combination matrix G MDS code minimum distance multiplicity nonzero nth root parameter parity check matrix perfect code permutation polynomial of degree positions Preparata code primitive element primitive nth root QR code quaternary code rational functions rational points reader Reed-Muller code Reed-Solomon code representation resp result ring root of unity rows of G Section sequence Show subcode symbols unique vector space weight enumerator words of weight yields zeros