曲面上2-型微分的留数代数及其在纠错码中的应用
Residus de 2-formes differentielles sur les surfaces algebriques et applications aux codes correcteurs d'erreurs
摘要：
代数-几何码的理论是在80年代初V.D.哥帕给定有限域上的光滑射影代数曲线X，有两种不同的纠错码构造方法。第一个，称为“泛函”，使用X上的一些有理函数，第二个，称为“微分”，涉及这个曲线上的一些有理1-形式。数百篇论文致力于研究这种代码。另外，Y.马宁在1984年的一篇文章中。关于这类码的一些论文已经发表，但是关于将微分构造推广到高维情形的工作还没有做。在这篇论文中，我们提出了代数曲面上码的微分构造。然后，我们研究了这些码的性质，特别是它们与函数码的关系。一个相当令人惊讶的事实是，与曲线情况的主要区别出现了。事实上，如果在曲线的情况下，微分代码总是函数代码的正交，那么这个断言通常对曲面失败。最后的观察激发了代码的研究，这些代码是表面上的一些功能代码的正交。因此，我们证明了，在一定条件下，表面上，这些码可以实现为差分码的和。此外，我们表明，一些开放的问题的答案“一拉贝尔蒂尼”可以给这些代码的参数非常有趣的信息。
英文摘要：
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.
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