Test your intuition: What is the simplest explanation you can give to the fact that the density of every packing of unit balls in is exponentially small in ?
Answers are most welcome.
Of course, understanding the asymptotic behavior of the density of densest packing of unit spheres in is a central problem in geometry. It is a long standing hope (perhaps naïve) that algebraic-geometry codes will eventually lead to examples showing that giving an exponential improvement of Minkowsky’s 1905 bound. (For more on sphere packing asymptotically and in dimensions 8 and 24 see this post.)
The result by Serge Vlăduţ from the previous post can be seen (optimistically) as a step in the direction of exponential improvement to Minkowski’s bound.