Algebraic and geometric approaches to lattices

Author(s)

Tatiana Bertoldi Carlos

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2007-09-05

Examining board members

Sueli Irene Rodrigues Costa; Antonio Aparecido de Andrade; José Plínio de Oliveira Santos; Paulo Roberto Brumatti; Reginaldo Palazzo Júnior

• Advisor: Sueli Irene Rodrigues Costa
• Field of knowledge: Physical Sciences and Mathematics - Mathematics
• Indexed in: Base Acervus-UNICAMP
• Location: Universidade Estadual de Campinas. Biblioteca Central Cesar Lattes; T/UNICAMP; C196a; Universidade Estadual de Campinas. Biblioteca do Intituto de Matemática, Estatística e Computação Científica; T/UNICAMP; C196a

Abstract

In this work we approach lattice constructions using properties of algebraic number theory. One focus is on the construction of ideal lattices via cyclotomic fields. Those lattices have been used as an efficient tool for designing coding strategies for the Rayleigh fading channels since it is possible to estimate the product distance and the diversity, parameters which control the error probability transmission for those channels. A special case, due to "shaping gain", is when those lattices are rotations of the n-dimensional integer lattice. We present a new construction of such lattices when n is a power of 2, via the maximal sub-field of the n-cyclotomic field. We also establish conditions for an ideal lattice to be a Zn-lattice using the Minkowski and the LLL (Lenstra-Lenstra-Lovasz) reductions. Other results include geometric characterizations of circulant graphs and of some algebraic lattices. (AU)