By Edgar Martinez-Moro, Carlos Munuera, Diego Ruano

"Advances in Algebraic Geometry Codes" offers the main profitable functions 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 data 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 recuperating the despatched details. With contributions shape popular researchers, this pioneering publication might be of price to mathematicians, desktop scientists, and engineers in info concept.

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1. Introduction The work on decoding of algebraic geometry codes started in 1986 and in the following 10 years a lot of papers appeared. The paper [11] surveys all the work on decoding until 1995. In this chapter we will present decoding algorithms using recent ideas and methods. 2 we present the basic algorithm for decoding a general algebraic geometry code CL (D, G), this algorithm only decodes error-patterns of weight smaller than d−1 2 − g where d is the Goppa bound on the minimum distance of the code and g is the genus of the curve used in the construction.

This is the case if deg G = n + 1 + 2g − 2 − 2deg G ≥ 4g + t, that is if 3t < n − 6g. The theorem shows that for a curve X /F of genus g with N rational points, and for 3t+4g < n ≤ N −1, there exist linear secret sharing schemes Σ = Σ0 (G, P) on n participants such that - Σ reject all subsets of size t, and - Σ reconstructs products of secrets from any n − t products of shares. One of the main results in [9] is that efficient linear secret sharing schemes for an increasing number of participants can be constructed over a small base field using asymptotically good families of curves.

Brouwer, and H. Wilbrink, Hermitian unitals are code words, Discrete Math. 97(1-3), 63–68, (1991). [7] C. Carvalho and F. Torres, On Goppa codes and Weierstrass gaps at several points, Des. Codes Cryptogr. 35(2), 211–225, (2005). -Y. Chen and I. M. Duursma, Geometric Reed-Solomon codes of length 64 and 65 over F8 , IEEE Trans. Inform. Theory. 49(5), 1351–1353, (2003). [9] H. Chen and R. Cramer. Algebraic geometric secret sharing schemes and secure multi-party computations over small fields. In CRYPTO, pp.