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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 theydo not exist for cardinality $2^n-5$.

Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$