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 profitable functions of algebraic geometry to the sphere of error-correcting codes, that are utilized in the whilst one sends info 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 therefore 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 conception.

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57. For a divisor G such that there exists a differential η with (η) = 3G − D − P0 , the AG-LSSS Σ0 (G, P) is trilinear. We give such a choice for the Hermitian curve X /F16 : Y 4 Z +Y Z = X 5 . 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.

Reine Angew. Math. 440, 99–109, (1993). [41] J. P. Hansen, Linkage and codes on complete intersections, Appl. Algebra Engrg. Comm. Comput. 14(3), 175–185, (2003). [42] J. W. P. Hirschfeld. Linear codes and algebraic curves. In Geometrical combinatorics (Milton Keynes, 1984), vol. 114, Res. , pp. 35–53. Pitman, Boston, MA, (1984). [43] J. W. P. Hirschfeld, G. Korchm´ aros, and F. Torres. The number of points on an algebraic curve over a finite field. In Surveys in combinatorics 2007, vol. 346, London Math.

The tables give the monomial basis for each of the two coordinate rings when q = 3. − 1 x x2 x3 x4 x5 x6 x7 x8 1 0 3 6 9 12 15 18 21 24 y 4 7 10 13 16 19 22 25 28 y2 8 11 14 17 20 23 26 29 32 − 1 x x2 x3 1 y y2 0 4 8 3 7 11 6 10 14 9 13 17 y3 12 15 18 21 y4 16 19 22 25 y5 20 23 26 29 y6 24 27 30 33 The coordinate ring F[x, y] of the Hermitian curve itself is often considered as an F[x] algebra with free basis {1, y, . . , y q−1 }. For one-point codes of full length q 3 + 1, the ring F[x, y] may be considered as an F[y] algebra with free basis {1, x, .

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