By Edgar Martinez-Moro, Carlos Munuera, Diego Ruano

"Advances in Algebraic Geometry Codes" provides the main profitable purposes of algebraic geometry to the sector 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 help (for example, compact disc). An error-correcting code therefore provides additional info to the message to be transmitted with the purpose of recuperating the despatched info. With contributions shape well known researchers, this pioneering booklet could be of worth to mathematicians, laptop scientists, and engineers in details concept.

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Mr are their pole orders then the set of all possible pole orders is the semigroup Λ = Zm1 + · · · + Zmr ⊂ Z The complement Z\Λ is finite of size g. Especially when r is small, the ring R can be used for efficient encoding (if the code is a onepoint code) or efficient decoding (using a key equation in standard form if the dual code is a one-point code, or a key equation in Welch-Berlekamp form if the code is a one-point code). M. Duursma The Hermitian curve over a field F of size q 2 is the curve X/F : y q + y = x .

3 = 1. In char = 3 the first tower X0 (2n ) is of type (D), and in char = 2 the second tower X0 (3n ) is of type (C). The equation Fn+1 = Fn (xn+1 ), x2i+1 + xi+1 = xi + 1 + 1/xi defines an asymptotically good tower over F8 . It has a generalization to arbitrary cubic fields. 4. One-point codes For a curve with many rational points for a given genus, any choice of divisor G will give a good code. In many cases, once the degree of G has been fixed, a convenient choice is a divisor G = mP∞ with support at a single point P∞ .

If at most t shares are corrupted then it is a priori possible to detect the corrupted shares and to determine their correct value. The assumption 4g + 2t = 2deg G < n − t that is used for schemes that are multiplicative in n − t positions corresponds to the much weaker deg G ≤ n − (2t + 1) − 2g. 10. Let (s1 , . . M. Duursma most t positions from the vector (f (P1 ), . . , f (Pn )), for f ∈ L(G). After choosing a suitable divisor F , we solve for g ∈ L(F ) and h ∈ L(G + F ) such that g(Pi )si = h(Pi ) for i = 1, .