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Coding Theory and Number Theory

  • Book
  • © 2003

Overview

Authors:
  1. Toyokazu Hiramatsu

    1. Hosei University, Tokyo, Japan

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  2. Günter Köhler

    1. Würzburg University, Würzburg, Germany

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Part of the book series: Mathematics and Its Applications (MAIA, volume 554-A)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xi
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  2. Linear Codes

    • Toyokazu Hiramatsu, Günter Köhler
    Pages 1-16

  3. Diophantine Equations and Cyclic Codes

    • Toyokazu Hiramatsu, Günter Köhler
    Pages 17-22

  4. Elliptic Curves, Hecke Operators and Weight Distribution of Codes

    • Toyokazu Hiramatsu, Günter Köhler
    Pages 23-48

  5. Algebraic-Geometric Codes and Modular Curve Codes

    • Toyokazu Hiramatsu, Günter Köhler
    Pages 49-75

  6. Theta Functions and Self-Dual Codes

    • Toyokazu Hiramatsu, Günter Köhler
    Pages 77-115

  7. Back Matter

    Pages 117-148
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Keywords

About this book

This book grew out of our lectures given in the Oberseminar on 'Cod­ ing Theory and Number Theory' at the Mathematics Institute of the Wiirzburg University in the Summer Semester, 2001. The coding the­ ory combines mathematical elegance and some engineering problems to an unusual degree. The major advantage of studying coding theory is the beauty of this particular combination of mathematics and engineering. In this book we wish to introduce some practical problems to the math­ ematician and to address these as an essential part of the development of modern number theory. The book consists of five chapters and an appendix. Chapter 1 may mostly be dropped from an introductory course of linear codes. In Chap­ ter 2 we discuss some relations between the number of solutions of a diagonal equation over finite fields and the weight distribution of cyclic codes. Chapter 3 begins by reviewing some basic facts from elliptic curves over finite fields and modular forms, and shows that the weight distribution of the Melas codes is represented by means of the trace of the Hecke operators acting on the space of cusp forms. Chapter 4 is a systematic study of the algebraic-geometric codes. For a long time, the study of algebraic curves over finite fields was the province of pure mathematicians. In the period 1977 - 1982, V. D. Goppa discovered an amazing connection between the theory of algebraic curves over fi­ nite fields and the theory of q-ary codes.

Authors and Affiliations

  • Hosei University, Tokyo, Japan

    Toyokazu Hiramatsu

  • Würzburg University, Würzburg, Germany

    Günter Köhler

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