Análise dos algoritmos de decodificação para códigos de geometria algébrica sobre curvas de hermite.

Abstract

On error-control coding, in communication systems, the most important development in the last years was the theory of algebraic geometric codes (AGC's), or geometric Goppa codes. This theory allows to get codes with better parameters, and forms an extremely elegant mathematical approach. Briefly, an ACG of length n consists in evaluating functions from a space of rational functions generated by a divisor G of an algebraic curve X, where this evaluation Is made over a set of n rational points of X disjoint from the support of G. ACG's based on hermitian curves have excellent parameters and are very used and referred in the literature, being the present work restricted to these codes. Its decoding has been made basically following two approaches: solving a set of linear equations over a location field, where syndromes are defined as a map from one linear subspace of rational functions to this location field, and solving a key equation in an affme ring, where syndromes are defined as elements in this affine ring. Skorobogatov and VlMu^'s basic algorithm and Porter's algorithm are typical examples of first and second approaches, respectively. AGGs' fast decoding, with smaller complexity (< O (n3)), has been got using Sakata's algorithm BMS, mostly with Feng and Rao's majority voting scheme. Ail of these schemes are described and analysed here.