## Algebraic Function Fields and Codes15 years after the ?rst printing of Algebraic Function Fields and Codes,the mathematics editors of Springer Verlag encouraged me to revise and extend the book. Besides numerous minor corrections and amendments, the second edition di?ers from the ?rst one in two respects. Firstly I have included a series of exercises at the end of each chapter. Some of these exercises are fairly easy and should help the reader to understand the basic concepts, others are more advanced and cover additional material. Secondly a new chapter titled “Asymptotic Bounds for the Number of Rational Places” has been added. This chapter contains a detailed presentation of the asymptotic theory of function ?elds over ?nite ?elds, including the explicit construction of some asymptotically good and optimal towers. Based on these towers, a complete and self-contained proof of the Tsfasman-Vladut-Zink Theorem is given. This theorem is perhaps the most beautiful application of function ?elds to coding theory. The codes which are constructed from algebraic function ?elds were ?rst introduced by V. D. Goppa. Accordingly I referred to them in the ?rst edition as geometric Goppa codes. Since this terminology has not generally been - cepted in the literature, I now use the more common term algebraic geometry codes or AG codes. I would like to thank Alp Bassa, Arnaldo Garcia, Cem Guneri, ̈ Sevan Harput and Alev Topuzo? glu for their help in preparing the second edition. |

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### Contents

1 | |

Algebraic Geometry Codes | 45 |

Extensions of Algebraic Function Fields | 67 |

Differentials of Algebraic Function Fields | 155 |

Algebraic Function Fields over Finite Constant Fields | 185 |

Examples of Algebraic Function Fields 217 | 216 |

Asymptotic Bounds for the Number of Rational Places | 243 |

More about Algebraic Geometry Codes 289 | 288 |

Subﬁeld Subcodes and Trace Codes | 311 |

Appendix A Field Theory | 327 |

Appendix B Algebraic Curves and Function Fields | 335 |

List of Notations | 345 |

References 349 | 348 |

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### Common terms and phrases

algebraic extension algebraic function ﬁeld algebraic geometry algebraically closed assume automorphism Bound canonical divisor charK Choose Cl(F coding theory consider constant ﬁeld extension Corollary cyclic deﬁned Deﬁnition degA denote differential divisor class element equation exists F I K(x F/IFq ﬁeld F ﬁeld of F ﬁeld of genus ﬁnd ﬁnite extension ﬁnite ﬁeld ﬁrst full constant ﬁeld function ﬁeld F/K G 9p G IFq Galois extension genus g Goppa codes hence IFqm IFqz integral integral closure irreducible isomorphism Lemma Let F/K minimal polynomial minimum distance obtain pairwise distinct place P G IPF places of degree places of F pole prime Proposition prove ramiﬁed rational function ﬁeld rational places residue class ﬁeld resp Riemann-Roch Theorem satisﬁes separable extension Show splits completely subﬁeld subgroup sufﬁcient Suppose Triangle Inequality unique unramiﬁed valuation ring vector space weakly ramiﬁed zero