 By Edgar Martinez-Moro, Carlos Munuera, Diego Ruano

ISBN-10: 981279400X

ISBN-13: 9789812794000

"Advances in Algebraic Geometry Codes" offers the main profitable purposes of algebraic geometry to the sphere of error-correcting codes, that are utilized in the while one sends details 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 info to the message to be transmitted with the purpose of getting better the despatched info. With contributions shape well known researchers, this pioneering ebook might be of worth to mathematicians, desktop scientists, and engineers in details concept.

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Sample text

Then the complement Q = D − Q ∼ (n − m∗ )P∞ + E. As in the first proof, Q + ER ∼ ((a0 + deg E)(q + 1) − a1)P∞ and a1 ≤ a0 + deg E. 42. Let A = B = (a0 (q + 1) − q)P∞ , Z = (q − a0 − 1)P∞ , 1 ≤ a0 ≤ q − 1. Then the code CΩ (G, D) with G = (2a0 −1)(q +1)−a0 has d = d∗ +(q −a0 −1). This corresponds to the case G = ((q − 2)(q + 1) + (2a0 + 1 − q)(q + 1) − a0 )P∞ in the theorem above, with a0 ≥ 2a0 + 1 − q if and only if a0 ≤ q − 1. For Hermitian one-point codes of length q 3 , the q 3 finite rational points form a complete intersection with coordinate ring 2 F[x, y]/(y q + y − xq+1 , xq − x) = xi y j : 0 ≤ i ≤ q 2 − 1, 0 ≤ j ≤ q − 1 .

97, Cambridge Tracts in Mathematics, (Cambridge University Press, Cambridge, 1991).  H. Niederreiter and C. P. Xing, Low-discrepancy sequences obtained from algebraic function fields over finite fields, Acta Arith. 72(3), 281–298, (1995).  H. Niederreiter and C. Xing, Rational points on curves over finite fields: theory and applications. vol. 285, London Mathematical Society Lecture Note Series, (Cambridge University Press, Cambridge, 2001).  S. Park, Applications of algebraic curves to cryptography, Thesis (University of Illinois, Urbana, 2007).

In Coding theory and algebraic geometry (Luminy, 1991), vol. , pp. 18–25. Springer, Berlin, (1992). D. Ehrhard, Achieving the designed error capacity in decoding algebraicgeometric codes, IEEE Trans. Inform. Theory. 39(3), 743–751, (1993). N. D. Elkies. Explicit modular towers. In Proceedings of the Thirty-Fifth Annual Allerton Conference on Communication, Control and Computing (Univ. of Illinois at Urbana-Champaign. N. D. Elkies. Excellent codes from modular curves. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp.