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Field name	Details
Dewey Class	004.0151
Title	Coding Theory and Number Theory ([EBook] /) / by Toyokazu Hiramatsu, Günter Köhler.
Author	Hiramatsu, Toyokazu
Added Personal Name	Köhler, Günter author.
Other name(s)	SpringerLink (Online service)
Publication	Dordrecht : : Springer Netherlands : : Imprint: Springer, , 2003.
Physical Details	XII, 148 p. : online resource.
Series	Mathematics and its applications ; 554-A
ISBN	9789401703055
Summary Note	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.:
Contents note	1. Linear Codes -- 2. Diophantine Equations and Cyclic Codes -- 3. Elliptic Curves, Hecke Operators and Weight Distribution of Codes -- 4. Algebraic-Geometric Codes and Modular Curve Codes -- 5. Theta Functions and Self-Dual Codes -- The Kloosterman Codes and Distribution of the Weights -- 1 Introduction -- 2 Melas code and Kloosterman sums -- 3 Hyper-Kloosterman code -- 4 Quasi-cyclic property -- 5 Weight distribution -- 7 A divisibility theorem for Hamming weights -- References.
System details note	Online access to this digital book is restricted to subscription institutions through IP address (only for SISSA internal users)
Internet Site	http://dx.doi.org/10.1007/978-94-017-0305-5
Links to Related Works	Subject References: Algebra . Algebraic Geometry . Coding and Information Theory . Coding theory . Computer Science . Discrete Mathematics in Computer Science . Linear and Multilinear Algebras, Matrix Theory . Matrix theory . Number Theory . Authors: Hiramatsu, Toyokazu . Köhler, Günter . Corporate Authors: SpringerLink (Online service) . Series: Mathematics and its applications . Classification: 004.0151 .

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Coding Theory and Number Theory

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