# Advances In Algebraic Geometry Codes by Edgar Martinez-Moro, Carlos Munuera, Diego Ruano PDF

By Edgar Martinez-Moro, Carlos Munuera, Diego Ruano

ISBN-10: 981279400X

ISBN-13: 9789812794000

"Advances in Algebraic Geometry Codes" offers the main winning functions of algebraic geometry to the sphere of error-correcting codes, that are utilized in the while one sends info via a loud channel. The noise in a channel is the corruption of part of the knowledge as a result of both interferences within the telecommunications or degradation of the information-storing aid (for example, compact disc). An error-correcting code hence provides additional details to the message to be transmitted with the purpose of improving the despatched info. With contributions shape well known researchers, this pioneering publication should be of worth to mathematicians, desktop scientists, and engineers in info conception.

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Ce cours de topologie a été dispensé en licence à l'Université de Rennes 1 de 1999 à 2002. Toutes les constructions permettant de parler de limite et de continuité sont d'abord dégagées, puis l'utilité de l. a. compacité pour ramener des problèmes de complexité infinie à l'étude d'un nombre fini de cas est explicitée.

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Example text

C23 , 0, . . , 0) ∈ CL (41P∞ , D) of weight 23. The nonzero positions lie on the lines 1 : x = α5 , 2 : x = α10 , 3 : y = (x + 1), 4 : y = α5 (x + 1), 5 : y = α10 (x + 1). Let Q1 = ( Q2 = 1 4 − P∞ ) + ( + 5 2 − P∞ ) + ( 3 − (0, 1)) ∼ 13P∞ − (0, 1). + (0, 1) ∼ 10P∞ + (0, 1). The vanishing ideals for Q1 and Q2 are generated by Q1 : (x5 + y 4 + y, f1 = x2 y + · · · , g1 = x6 + · · · ), Q2 : (x5 + y 4 + y, f2 = x2 y 2 + · · · , g2 = y 3 + · · · ). algebraic August 25, 2008 10:59 World Scientific Review Volume - 9in x 6in algebraic 37 Algebraic Geometry Codes: General Theory − 1 x x2 x3 x4 1 − − − − − y y2 −− −− ++ ++ ++ y3 − − + + + y4 − + + + + y5 + + + + + y6 + + + + + ··· ··· ··· ··· ··· ··· − 1 x x2 x3 x4 1 y −− −− −− −− −− y2 − + + + + y3 + + + + + y4 + + + + + y5 + + + + + y6 + + + + + ··· ··· ··· ··· ··· ··· If the word c = (c1 , .

Pn . In that case CΩ (mP∞ , D) = CL ((2g − 2 + n − m)P∞ , D). For the projective line, for the Hermitian curves, and for the Suzuki curves, the dual of a one-point code is again a one-point code. For each of these curves, the divisor D can be chosen to be the set of all rational points minus the point P∞ . For this choice of D, there exists an algebraic function x ∈ K such that n = [K : F(x)] · q and η = df /f for f = xq − x.. The one-point codes can be extended by including the point P∞ in D.

V. Kumar, H. Stichtenoth, and V. Deolaikar, A low-complexity algorithm for the construction of algebraic-geometric codes better than the Gilbert-Varshamov bound, IEEE Trans. Inform. Theory. 47 (6), 2225–2241, (2001). S. A. Stepanov, Codes on algebraic curves. (Kluwer Academic/Plenum Publishers, New York, 1999). H. Stichtenoth, Algebraic function fields and codes. Universitext, (SpringerVerlag, Berlin, 1993). H. Stichtenoth and C. Xing, Excellent nonlinear codes from algebraic function fields, IEEE Trans.