Gittern建筑工程
Algorithmische Konstruktionen von Gittern
摘要：
本文的主要目的是对整格进行分类。利用Kneser近邻法，我们编制了一个计算机程序tn，对整格的完备属进行分类。主要结果是详细的分类模格的维度高达14的水平3，5，7，和11。我们提出了一个快速的元算法计算的基础格，这是由一个大的生成系统。理论上的最坏情况边界和实际的实验表明，与传统的方法相比，重要的优势。小的修改，我们使用这个算法的分解成成对正交子格。作为进一步的结果，我们证明了格的生成系统的覆盖定理。设S是秩为n的格的极小生成系，其生成元不大于B。则S至多有n + log_2（n!（B/M）^n）元素。我们描述和计算格L的不变量，就像L/2L上的长度函数的离散傅里叶变换的谱一样。讨论了格量化器和格陪集并量化器的数值计算。
英文摘要：
The main objective of this thesis is a classification project for integral lattices. Using Kneser's neighbour method we have developed the computer program tn to classify complete genera of integral lattices. Main results are detailed classifications of modular lattices in dimensions up to 14 with levels 3,5,7, and 11. We present a fast meta algorithm for the computation of a basis of a lattice which is given by a large generating system. A theoretical worst case boundary and practical experiments show important advantages in comparison to traditional methods. Up to small modifications we use this algorithm for the decomposition of a lattice into pairwise orthogonal sublattices. As a further result we prove a covering theorem for the generating system of a lattice. Let S be a minimal generating system of a lattice of rank n with generators not larger than B. Then S has at most n + log_2(n!(B/M)^n) elements. We describe and compute invariants of a lattice L like the spectrum of the discrete Fourier transform of the length function on L/2L. The numerical computation of a lattice quantizer and a quantizer of a union of cosets of a lattice are discussed.
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