arXiv 1304.21452013-04-08math.NT

渐近最优函数域塔的自同构与子塔

On Automorphisms and Subtowers of an asymptotically optimal Tower of Function Fields

Thorsten Lagemann

摘要：

本文研究了由Garcia和Stichtenoth引入的函数域的渐近最优塔的自同构群。特别是我们提供了一些合理的地方的分解群的详细描述。这个群作用于由Garcia-Stichtenoth塔得到的代数几何标准码，超过Gilbert-Varshamov界。由分解群固定的域形成了一个渐近最优的非伽罗瓦子塔，这是Bezerra和Garcia首先发现的，并改进了计算Gilbert-Varshamov界以上的代码。本文还描述了它与Garcia-Stichtenoth塔的比例关系，并得到了它的有理空间及其Weierstraße半群的新的精确结果。

英文摘要：

In this article we investigate the automorphism group of an asymptotically optimal tower of function fields introduced by Garcia and Stichtenoth. In particular we provide a detailed description of the decomposition group of some rational places. This group acts on the algebraic-geometric standard codes obtained by the Garcia-Stichtenoth tower exceeding the Gilbert-Varshamov bound. The fields fixed by the decomposition groups form an asymptotically optimal non-Galois subtower, which has been first found by Bezerra and Garcia and yields an improvement for computing codes above the Gilbert-Varshamov bound. In this article we also describe its proportionality to the Garcia-Stichtenoth tower and obtain new precise results on its rational places and their Weierstraß semigroups.

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