Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Bücher
Zeitschriften
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
MTZ-Fachkongress

Antriebe und Energiesysteme von morgen


Austausch von Automobil- und Energiewirtschaft

Am 14./15. Mai geht es in Chemnitz um die Mobilitätswende durch Elektro- und Wasserstofffahrzeuge.
Erweiterte Suche
Anmelden
nach oben

1999 | Buch

Codes and Their Parameters Kapitel lesen Erstes Kapitel lesen

Codes on Algebraic Curves

verfasst von: Serguei A. Stepanov

Verlag: Springer US

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten
insite
SUCHEN

Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Error-Correcting Codes

Frontmatter
Chapter 1. Codes and Their Parameters
Abstract
In 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.
Serguei A. Stepanov
Chapter 2. Bounds on Codes
Abstract
We have already explained that a good code should have large d/n andk/nin the unit interval [0,1] for a givenn.From Shannon’s theorem we know also that we should study long codes. However, if the channel has symbol-error probabilitypthen we should expect an average ofpnerrors per received word of lengthn.To correct these we need to have a minimum distance more than2pn.So, if we increasenthendshould increase proportionally.
Serguei A. Stepanov
Chapter 3. Examples and Constructions
Abstract
We now turn to the problem of constructing linear codes. We present several examples, each of which is in fact a method to construct some family of linear codes having rather good parameters. A considerable part of these families are predecessors of geometric Goppa codes, therefore we treat the corresponding constructions in this way to demonstrate their close interconnection with the Goppa construction that will be described later on in Part IV.
Serguei A. Stepanov

Algebraic Curves and Varieties

Frontmatter
Chapter 4. Algebraic Curves
Abstract
This chapter contains the basic definitions and results of the theory of algebraic curves: valuations, divisors, the genus of a curve, finite morphisms, linear systems, Jacobians, differential forms and their residues, the Riemann—Roch theorem, Hurwitz and Plücker genus formulas, special divisors and Weierstrass points. We do not consider here the arithmetical properties of curves and for that reason the ground fieldkis assumed to be algebraically closed.
Serguei A. Stepanov
Chapter 5. Curves over a Finite Field
Abstract
In Chapter 4 we have assumed that the ground fieldkis algebraically closed. However, if we are interested in the consideration of arithmetic properties of algebraic varieties, we must develop the corresponding theory for the case of non-closed fields such as ℚ or F q .For example, in applying algebraic geometry to coding theory, one should study curves defined over F q and their points with coordinates in F q (such points are called F q -rational).
Serguei A. Stepanov
Chapter 6. Counting Points on Curves over Finite Fields
Abstract
In this chapter we apply the technique we have worked out earlier to prove the Riemann hypothesis for the zeta-function ζ (X, s) of a curve X defined over a finite field F q .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 X. Here we give an elementary proof based essentially on using only the Riemann—Roch theorem (see Stepanov [184, 185, 187], Bombieri [17], Schmidt [159] and Stöhr and Voloch [200]).
Serguei A. Stepanov

Elliptic and Modular Curves

Frontmatter
Chapter 7. Elliptic Curves
Abstract
The theory of elliptic curves (curves of genus 1 having a specified basepoint xo) is varied and rich, and provides a good example of the profound connections between abstract algebraic geometry, complex analysis, and number theory. The most important property is that any elliptic curve is an abelian variety.
Serguei A. Stepanov
Chapter 8. Classical Modular Curves
Abstract
This chapter contains an analytical description of classical modular curves and introduces the Hecke theory of modular forms which later on will be used for the study of arithmetical properties of modular curves of a special form.
Serguei A. Stepanov
Chapter 9. Reductions of Modular Curves
Abstract
The existence of good codes coming from classical modular curves is substantiated by the following three phenomena:
(i)
the existence of modular curves,i.e.,curves whose points have an interpretation as modular points;
 
(ii)
the zeta-function of such curves overF p is expressible in terms of Fourier coefficients of normalized eigenforms for the algebra of Hecke operators;
 
(iii)
the Eichler-Selberg trace formula,which computes the trace of a Hecke operator on the space of modular forms.
 
Serguei A. Stepanov

Geometric Goppa Codes

Frontmatter
Chapter 10. Constructions and Properties
Abstract
This chapter describes Goppa’s construction of linear error-correcting codes coming from algebraic curves over finite fields. There exist several essentially equivalent ways to construct such codes. The first to be considered is the so-called L-construction.
Serguei A. Stepanov
Chapter 11. Examples
Abstract
In this chapter we describe several examples of geometric Goppa [n, k, d]q-codes coming from various algebraic curves defined over a finite field F q .
Serguei A. Stepanov
Chapter 12. Decoding Geometric Goppa Codes
Abstract
This chapter concerns the decoding problem for geometric Goppa codes. We consider various aspects of the problem beginning with results on the existence 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].
Serguei A. Stepanov
Chapter 13. Bounds
Abstract
The significance of geometric Goppa codes is clarified when we consider asymptotic problems.
Serguei A. Stepanov
Backmatter
Metadaten
Titel
Codes on Algebraic Curves
verfasst von
Serguei A. Stepanov
Copyright-Jahr
1999
Verlag
Springer US
Electronic ISBN
978-1-4615-4785-3
Print ISBN
978-1-4613-7167-0
DOI
https://doi.org/10.1007/978-1-4615-4785-3