In the summer of 1989, Voyager II sent images of Neptune before heading out of the solar system. The transmitted 24-bit codewords were designed to be the centers of non-overlapping spheres in 24 dimensional space, hence the noisy message received on earth was perfectly recovered for codewords that remained within their dedicated spheres. The channel code efficiency for this design is mainly a matter of how densely one could pack the spheres, lucky for Voyager 2, packing on Leech lattice (discovered in 1967) was known to be dense, although it was proved to be the densest packing only in 2016, when the spacecraft was 17000 million kilometers away. In addition to the 24 dimensional Euclidean space, densest sphere packing is known in one (trivial), two, three and eight dimensional spaces. Very little is known, for the other dimensions, and in most cases, even an intuition is lacking. Even in the aforementioned dimensions, packing with arbitrary objects is not well understood.

One of the applications of my research is to provide numerical bounds and constructions for dense packing of arbitrary convex bodies in N-dimensional spaces. A generalizable upper-bound for the density of packing can be expressed as Linear Optimizations with Fourier-analytic constraints. The problem of packing is notoriously difficult and have long been only a subject of theoretical approaches in pure mathematics. We introduced a computational approach to provide approximations, insights and intuitions for the solution of this problems.

[github] [slides]