Problem
I need to create 32 Bit numbers (signed or unsigned doesn't matter, the highest bit will never be set anyway) and each number must have a given number of Bits set.
Naive Solution
The easiest solution is of course to start with the number of zero. Within a loop the number is now increased by one, the number of Bits is counted, if the count has the desired value, the number is stored to a list, if not the loop just repeats. The loop is stopped if enough numbers have been found. Of course this works just fine, but it's awfully slow once the number of desired Bits gets very high.
A Better Solution
The simplest number having (let's say) 5 Bits set is the number where the first 5 Bit are set. This number can be easily created. Within a loop the first bit is set and the number is shifted to the left by one. This loop runs 5 times and I found the first number with 5 Bits set. The next couple of numbers are easy to create as well. We now pretend the number to be 6 Bit wide and the highest one is not set. Now we start shifting the first zero bit to the right, so we get 101111, 110111, 111011, 111101, 111110. We could repeat this by adding another 0 to front and repeating this process. 0111110, 1011110, 1101110, etc. However that way numbers will grow much faster than necessary, as using this simple approach we leave out numbers like 1010111.
So is there a better way to create all possible permutations, a generic approach, that can be used, regardless how many bits the next number will have and regardless how many set bits we need set?