Algorithmische Konstruktionen von Gittern

Abstract

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.