Lattices: an Algebraic Number Theory approach

• Grant number: 21/01326-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: May 01, 2021
• End date: April 30, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Agnaldo José Ferrari
• Grantee: Vitor Fróes Berbel
• Host Institution: Faculdade de Ciências (FC). Universidade Estadual Paulista (UNESP). Campus de Bauru. Bauru , SP, Brazil

Abstract

Algebraic number theory has played an important role in building codes and algebraic lattices. Find algebraic lattices via number fields with maximum minimum product distance has been the object of study in recent years. Algebraic lattices are those obtained from ring of integers of a number field and ideal lattices are algebraic endowed with a trace form. The ideal lattice theory has proved to be useful in information theory. Ideal lattices with high density packing have been studied as an alternative approach for transmitting signals to the Gaussian channel, which is a AWGN communication channel (Additive White Gaussian Noise), where prevail attenuations and delay in signal propagation. Ideal lattices with high diversity and minimum product distance are interesting for the transmission of signals to the Rayleigh fading channel, which is a communication channel that has as main characteristic the propagation by multiple paths. This project aims to: (I) Introduce the basic concepts related to lattices and spherical packing. (ii) Study the concepts related to algebraic number theory.(III) Applying the concepts seen in previous items in the construction of algebraic lattices and ideal lattices, reproducing some of the lattices known in the literature which may represent signal constellations which are efficient to both communication channels mentioned above.