aalto1 untyped-item.component.html
Lattices and packings of convex bodies
Loading...
URL
Journal Title
Journal ISSN
Volume Title
School of Science |
Doctoral thesis (article-based)
| Defence date: 2025-09-12
Electronic archive copy is available via Aalto Thesis Database.
Unless otherwise stated, all rights belong to the author. You may download, display and print this publication for Your own personal use. Commercial use is prohibited.
Authors
Date
Major/Subject
Mcode
Degree programme
Language
en
Pages
86 + app. 84
Series
Aalto University publication series Doctoral Theses, 147/2025
Abstract
The mainaim of this thesis is to develop tools needed to attack the lattice packing problem for a convex body. This is done by extending the existing theory of positive definite real quadratic forms to a theory of admissible lattices for a general convex body. Our search for dense lattice packings of convex bodies is motivated by the code design problem in a Rayleigh fading wiretap channel. Indeed, we show that a legitimate receiver's error probability and a malicious receiver's correct decoding probability are both controlled by the l1-theta series of code lattices.
Additionally, this thesis makes progress towards the kissing number problem for the cross-polytope by proving new asymptotic bounds for the lattice and translative kissing numbers, and by determining the lattice kissing number of the cross-polytope in dimension four. Furthermore, the asymptotic singularity probability of a random circulant Bernoulli matrix of order n is determined for all n, and new non-existence results for abelian difference sets and their multiplier groups are proved. Finally, the concept of arithmetically equivalent number fields is extended to central simple algebras over number fields.
Huvudsyftet med detta arbete är att utveckla metoder med vars hjälp det är möjligt att hitta gitterpackningar av konvexa kroppar med hög packningsdensitet. Packingsproblemet motiveras av koddesignproblemet för en Rayleigh fading avlyssningskanal. Vi visar att en legitim mottagares sannolikhet för avkodningsfel och en icke-legitim mottagares sannolikhet för korrekt avkodning beror på kodgittrenas l1-thetaserie.
Vi bestämmer asymptotiska gränser för kysstalet för en korspolytop i högdimensionell Euklidisk rymd samt bevisar att gitterkysstalet för en fyrdimensionell korspolytop är 40. Utöver detta så bestämmer vi den asymptotiska sannolikheten att en cyklisk matris är singulär, och härleder ickeexistensresultat för differensmängder i abelska grupper samt deras multiplikatorgrupper. Vi utvidgar konceptet om aritmetisk ekvivalens i en talkropp till centrala enkla algebror över talkroppar.
Description
Supervising professor
Hollanti, Camilla, Prof., Aalto University, Department of Mathematics and Systems Analysis, FinlandThesis advisor
Mantilla-Soler, Guillermo, Prof., Universidad Nacional de Colombia, ColombiaOther note
Parts
- [Publication 1]: Niklas Miller. A Design Criterion for the Rayleigh Fading Wiretap Channel Based on ℓ1-norm Theta Functions. Accepted for publication in SIAM Journal on Applied Algebra and Geometry, March 2025.
DOI: 10.48550/arXiv.2405.04143 View at publisher
- [Publication 2]: Niklas Miller. On the Kissing Number of the Cross-Polytope. Submitted, February 2025.
DOI: 10.48550/arXiv.2501.09245 View at publisher
- [Publication 3]: Niklas Miller. On the Singularity Probability of Random Circulant Bernoulli Matrices. Submitted, January 2025.
DOI: 10.48550/arXiv.2411.17577 View at publisher
- [Publication 4]: Niklas Miller. Forbidden Multipliers in Abelian Difference Sets. Submitted, May 2025
- [Publication 5]: Guillermo Mantilla-Soler and Niklas Miller. Arithmetic Equivalence for Central Simple Algebras over Number Fields. Submitted, May 2025