  It is known that the $n$-qubit system has no unextendible product bases
(UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the
$n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge3$. We prove that they
do not exist for cardinality $2^n-5$.
