## Codes on Algebraic CurvesThis is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A. |

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### Contents

I | 1 |

II | 3 |

III | 25 |

IV | 41 |

V | 69 |

VI | 71 |

VII | 103 |

VIII | 143 |

X | 175 |

XI | 193 |

XII | 219 |

XIII | 241 |

XIV | 243 |

XV | 257 |

XVI | 289 |

XVII | 315 |

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### Common terms and phrases

abelian additive character affine curve algebraic curves algebraically closed asymptotically automorphism basic algorithm bound called character of Fq classical modular curves code-vector coefficients complex congruence subgroup consider construction coordinates Corollary corresponding curve defined cyclic degD denote differential form discrete valuation divisor class divisor of degree effective divisor elements elliptic curve equation equivalence class error-vector exists finite field finite field Fq following result Fq-rational points genus formula genus g geometric Goppa codes Goppa codes Hecke operator hence homomorphism irreducible isomorphic k-rational lattice Lemma linear codes linear n,k,d]q-code minimum distance modular curves modular points modulo morphism multiplicative non-singular non-trivial non-zero obtain parameters parity-check matrix pole polynomial positive integer prime divisor projective curve Proof Proposition Prove ramified rational divisor rational function rational points resp Riemann-Roch theorem Show smooth projective curve subset supersingular SuppDo syndromes valuation ring Weierstrass Xo(N zero