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?

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 NOT optimal?