  We investigate the problem of constructing unextendible product bases in the
qubit case - that is, when each local dimension equals 2. The cardinality of
the smallest unextendible product basis is known in all qubit cases except when
the number of parties is a multiple of 4 greater than 4 itself. We construct
small unextendible product bases in all of the remaining open cases, and we use
graph theory techniques to produce a computer-assisted proof that our
constructions are indeed the smallest possible.
