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Algebraic construction of lattices via Minkowski's homomorphism

Grant number: 18/05324-2
Support type:Scholarships in Brazil - Scientific Initiation
Effective date (Start): June 01, 2018
Effective date (End): December 31, 2018
Field of knowledge:Physical Sciences and Mathematics - Mathematics
Principal Investigator:Agnaldo José Ferrari
Grantee:Otávio Benicio Mirandola
Home 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.