>>13823477This is a crude approach, but because the positions in the bitstrings are independent for the xor operation, you should be able to build up the 12 bitstrings one bit at a time, adding new bits to these 12 bitstrings until you have 12 nonzero, unique bitstrings.
So if you solve the xor relation for 12 bitstrings of length 1 (obviously not all 0-bits or all 1-bits), you should be able to solve the problem simply by shifting this length-1 relation isomorphically around the graph.
