Lattices, codes, and sphere packings

The sphere packing problem asks for the highest proportion of space that can be filled with non overlapping spheres of a same radius. Euclidean lattices, infinite analogues of linear codes, induce structured arrangements that turn out to be good packings. 

After a brief introduction to lattices and an overview about the known results in low dimension, we will focus on lattice sphere packings in high dimension. We will see how we can construct the densest known lattices by using codes defined over the ring of integers of cyclotomic fields, which yields an effective version of a result by Venkatesh. 

In the last part of the talk, we will discuss other problems involving lattices, codes and spheres, such as kissing numbers and spherical codes. We will try to give an idea on how semidefinite optimization techniques can be applied to solve these problems.