Brian Kurkoski (Japan Advanced Institute of Science and Technology)
Lattices are error-correcting codes defined over the real numbers. In the physical world, 1 plus 1 is 2, and it is the same for lattices. An important example of a physical-world application is wireless communications. Two electromagnetic signals, transmitted at the same time, will superimpose --- that is, they add, making lattice codes a natural fit for wireless communications. This lecture gives a brief introduction to lattices for those already familiar with the fundamentals of coding theory. Similarities with finite-field codes, such as minimum distance, matrix representation and constructions are emphasized, while unique aspects of lattices such as scaling, Voronoi regions and nested lattice codes are explained. Some recent trends and open problems in lattice coding will be reviewed.