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

By Edgar Martinez-Moro, Carlos Munuera, Diego Ruano

"Advances in Algebraic Geometry Codes" offers the main winning functions of algebraic geometry to the sector of error-correcting codes, that are utilized in the whilst one sends details via a loud channel. The noise in a channel is the corruption of part of the knowledge because of both interferences within the telecommunications or degradation of the information-storing aid (for example, compact disc). An error-correcting code hence provides additional info to the message to be transmitted with the purpose of getting better the despatched info. With contributions shape popular researchers, this pioneering e-book may be of worth to mathematicians, machine scientists, and engineers in details thought.

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It has 65 rational points that form a complete intersection X = P with aˆ defined with the Reed-Muller invariant a = q − 1 = 15. The LSSS Σ(C) ˆ code C = RM (ν = 5, X = P) is trilinear. The Reed-Muller code is equivalent to a geometric Goppa code defined with a divisor G ∼ 5L. The curve has parameters N = 65 and g = 6, the code Cˆ is of type [65, 20, 40], and ˆ has parameters n = 64 and t = 13. the scheme Σ(C) For a LSSS Σ0 (G, P) with deg G ≤ n − (2t + 1), any two vectors of shares differ in at least 2t + 1 positions.

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. Inform. Theory. 51(11), 4044–4046, (2005).

4 The generalized order bound . . . . . . . . . 5 Majority voting . . . . . . . . . . . . . 6 List decoding of algebraic geometry codes . . . . . . 7 Syndrome formulation of list decoding . . . . . . . 8 Literature . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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