Public defence in the field of Mathematics and Statistics, M.Sc. (Tech.) Niklas Miller

Title of the thesis: Lattices and Packings of Convex Bodies

Thesis defender: Niklas Miller 
Opponent: Prof. Padraig Ó Catháin, Dublin City University, Ireland
Custos: Prof. Camilla Hollanti, Aalto University School of Science

This thesis investigates the lattice packing problem for a convex body, proves new results about multiplier groups in abelian difference sets, and proves results about arithmetic equivalence in simple algebras. Asymptotic upper and lower bounds are derived for various lattice invariants such as the l^1-kissing number. It is proved, that the lattice D_4^+ solves the l^1-lattice kissing number problem in dimension four.

The results of the thesis can be applied in wireless communications: it is shown that in a wiretap channel with Rayleigh fading, an eavesdropper's correct decoding probability is minimzed, when the code lattices are chosen such that the l^1-theta series is minimized. The densest lattice packing minimizes the l^1-theta series asymptotically. We develop a method with which one can find a dense lattice packing of a given polytope and derive necessary conditions that a locally optimal lattice must satisfy.

Keywords: Lattices, combinatorial design theory, arithmetic equivalence, random matrix theory

Thesis available for public display 7 days prior to the defence at Aaltodoc.