Above dimension 3, the best sphere packing bounds known are all proved using some form of linear programming bounds. However, these bounds behave quite mysteriously. They appear to be sharp in 8 and 24 dimensions, yet no proof is known, despite tantalizing numerical clues. The asymptotic behavior of the bounds as the dimension tends to infinity also remains unclear. Furthermore, numerical calculations show unexpected and puzzling behavior.
In this talk, we'll explore what we don't understand about sphere packing bounds.