  Unextendible product bases have been shown to have many important uses in
quantum information theory, particularly in the qubit case. However, very
little is known about their mathematical structure beyond three qubits. We
present several new results about qubit unextendible product bases, including a
complete characterization of all four-qubit unextendible product bases, which
we show there are exactly 1446 of. We also show that there exist p-qubit UPBs
of almost all sizes less than $2^p$.
