Residus de 2-formes differentielles sur les surfaces algebriques et applications aux codes correcteurs d-erreursReport as inadecuate



 Residus de 2-formes differentielles sur les surfaces algebriques et applications aux codes correcteurs d-erreurs


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The theory of algebraic-geometric codes has been developed in the beginning of the 80-s after a paper of V.D. Goppa. Given a smooth projective algebraic curve X over a finite field, there are two different constructions of error-correcting codes. The first one, called -functional-, uses some rational functions on X and the second one, called -differential-, involves some rational 1-forms on this curve. Hundreds of papers are devoted to the study of such codes. In addition, a generalization of the functional construction for algebraic varieties of arbitrary dimension is given by Y. Manin in an article of 1984. A few papers about such codes has been published, but nothing has been done concerning a generalization of the differential construction to the higher-dimensional case. In this thesis, we propose a differential construction of codes on algebraic surfaces. Afterwards, we study the properties of these codes and particularly their relations with functional codes. A pretty surprising fact is that a main difference with the case of curves appears. Indeed, if in the case of curves, a differential code is always the orthogonal of a functional one, this assertion generally fails for surfaces. Last observation motivates the study of codes which are the orthogonal of some functional code on a surface. Therefore, we prove that, under some condition on the surface, these codes can be realized as sums of differential codes. Moreover, we show that some answers to some open problems -a la Bertini- could give very interesting informations on the parameters of these codes.



Author: A. Couvreur

Source: https://archive.org/







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