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＜電子ブック＞
Coding Theory and Number Theory / by T. Hiramatsu, Günter Köhler
(Mathematics and Its Applications ; 554-A)

版	1st ed. 2003.
出版者	Dordrecht : Springer Netherlands : Imprint: Springer
出版年	2003
本文言語	英語
大きさ	XII, 148 p : online resource
冊子体	Coding theory and number theory / by Toyokazu Hiramatsu and Gunter Kohler
著者標目	*Hiramatsu, T author Köhler, Günter author SpringerLink (Online service)
件　名	LCSH:Computer science -- Mathematics  全ての件名で検索 LCSH:Discrete mathematics LCSH:Algebraic geometry LCSH:Number theory LCSH:Coding theory LCSH:Information theory LCSH:Algebras, Linear FREE:Discrete Mathematics in Computer Science FREE:Algebraic Geometry FREE:Number Theory FREE:Coding and Information Theory FREE:Linear Algebra
一般注記	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 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 Accessibility summary: This PDF is not accessible. It is based on scanned pages and does not support features such as screen reader compatibility or described non-text content (images, graphs etc). However, it likely supports searchable and selectable text based on OCR (Optical Character Recognition). Users with accessibility needs may not be able to use this content effectively. Please contact us at accessibilitysupport@springernature.com if you require assistance or an alternative format Inaccessible, or known limited accessibility No reading system accessibility options actively disabled Publisher contact for further accessibility information: accessibilitysupport@springernature.com HTTP:URL=https://doi.org/10.1007/978-94-017-0305-5

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