in reply to
Re^2: decomposing binary matrices
in thread decomposing binary matrices
Sticking with the bitstring concept, I think rather that you are looking for a subset of n bitstrings such that the bitwise-OR of the bitstrings has only n distinct bits set.
Note that some or all of those bitstrings may have fewer than n bits set; you may even have two bitstrings that have no bit set in common in a qualifying subset (though you'd need a minimum of 4 elements in the subset to avoid it being further decomposable).