I am solving a problem where a set of N K-bit integers is given (N <= 10^4, K <= 50). I am allowed to do a single operation: choose two numbers A,B from the set and insert A xor B into the set.

The problem asks if it is possible, for the given set, to generate all numbers from 0 to 2^K - 1 using only this operation, as many times as i want.

I read that this can be solved using Gaussian Elimination, but i don't know how to do it. Can any one help me with this problem? Thx in advance :)