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＜電子ブック＞
Codes on Algebraic Curves / by Serguei A. Stepanov

版	1st ed. 1999.
出版者	New York, NY : Springer US : Imprint: Springer
出版年	1999
本文言語	英語
大きさ	XIII, 350 p : online resource
冊子体	Codes on algebraic curves / Serguei A. Stepanov
著者標目	*Stepanov, Serguei A author SpringerLink (Online service)
件　名	LCSH:Algebra LCSH:Electrical engineering LCSH:Algebraic geometry LCSH:Algorithms FREE:Algebra FREE:Electrical and Electronic Engineering FREE:Algebraic Geometry FREE:Algorithms
一般注記	I. Error-Correcting Codes -- 1 Codes and Their Parameters -- 2 Bounds on Codes -- 3 Examples and Constructions -- II. Algebraic Curves and Varieties -- 4 Algebraic Curves -- 5 Curves over a Finite Field -- 6 Counting Points on Curves over Finite Fields -- III. Elliptic and Modular Curves -- 7 Elliptic Curves -- 8 Classical Modular Curves -- 9 Reductions of Modular Curves -- IV. Geometric Goppa Codes -- 10 Constructions and Properties -- 11 Examples -- 12 Decoding Geometric Goppa Codes -- 13 Bounds -- List of Notations This 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 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/10.1007/978-1-4615-4785-3

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