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Titlebook: Codes on Algebraic Curves; Serguei A. Stepanov Book 1999 Kluwer Academic/Plenum Publishers 1999 Prime.Prime number.algebra.algebraic curve

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书目名称Codes on Algebraic Curves
编辑Serguei A. Stepanov
视频videohttp://file.papertrans.cn/229/228843/228843.mp4
图书封面Titlebook: Codes on Algebraic Curves;  Serguei A. Stepanov Book 1999 Kluwer Academic/Plenum Publishers 1999 Prime.Prime number.algebra.algebraic curve
描述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. Tsf
出版日期Book 1999
关键词Prime; Prime number; algebra; algebraic curve; algebraic geometry; algebraic varieties; algorithms; coding
版次1
doihttps://doi.org/10.1007/978-1-4615-4785-3
isbn_softcover978-1-4613-7167-0
isbn_ebook978-1-4615-4785-3
copyrightKluwer Academic/Plenum Publishers 1999
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Counting Points on Curves over Finite Fieldsld ..This result was proved for the first time by Hasse (in the case of elliptic curves) and Weil (in the general case) using the correspondence theory on .. Here we give an elementary proof based essentially on using only the Riemann—Roch theorem (see Stepanov [184, 185, 187], Bombieri [17], Schmid
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Decoding Geometric Goppa Codesce of decoding algorithms and ending with ones on the construction of efficient algorithms which can easily be used in practice. For a detailed treatment of the complexity of algorithms we refer the reader to Aho, Hoperoft and Ulman [2].
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https://doi.org/10.1007/978-3-030-53149-2In this chapter the basic notions of the theory of error-correcting codes are introduced: the Hamming distance, parameters of codes, linear codes, encoding and decoding procedures, spectrum and duality, the MacWilliams identity and Krawtchouk polynomials.
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