Covering Sets

Let's consider a set S of N different elements numbered from 0 to N - 1. It's a well-known fact that there are exactly 2N subsets of S (including the empty subset). Each subset S' of S can be encoded as a bitmask B(S') containing N bits, where the i'th bit of B(S') is equal to 1 if the i'th element of S belongs to S', and 0 otherwise. Each bitmask can also be considered as a non-negative integer r