There are some natural packing problems that have been asked in mathematics. Some of them are:

1)How many balls can be placed with in a cube?

2)How many equidistant points can be place on the surface of a sphere?

3)How many code points can one have asymptotically for a length n code with minimum distance(itself defined in various ways) d over an alphabet of size q?

There are many other generalizations to packing in spaces of different characteristics.

My question is given a packing, what are some of the most useful (avoiding exhaustive search) techniques available to show that packing is optimal or not?