WELL-ROUNDED LATTICES VIA QUADRATIC NUMBER FIEDLS

• Grant number: 23/05215-7
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2023
• End date: December 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Algebra
• Principal Investigator: Grasiele Cristiane Jorge
• Grantee: Leonardo Farias Santos
• Host Institution: Instituto de Ciência e Tecnologia (ICT). Universidade Federal de São Paulo (UNIFESP). Campus São José dos Campos. São José dos Campos , SP, Brazil

Abstract

A lattice is a discrete additive subgroup of R^n. It can be shown that given a lattice Lambda em R^n, there are m linearly independent vectors over R so that Lambda can be described as a linear combination of these m vectors with integer coefficients. A full rank lattice, that is, when m=n, is said to be well-rounded if the set consisting of its minimum Euclidean norm vectors generates R^n. Well-rounded lattices have been considered in the Error Correcting Code Theory for Wiretap Gaussian channels with multi-input, multiple-output (MIMO) and single-input, single-output (SISO). Recent works have studied the relationship between well-rounded lattices and algebraic lattices. A lattice in R^n is said to be algebraic if it can be obtained as the image of a canonical or twisted homomorphism applied to a free Z-module of rank n contained in a number field of degree n. Algebraic lattice constructions may be used to calculate some lattice parameters that are difficult to be calculated in general lattices in R^n. In this undergraduate research project we will focus on the study of which well-rounded lattices can be obtained via quadratic fields.