Multiqubit UPB: The method of formally orthogonal matrices

We use formal matrices whose entries we view as vector variables taking unit vectors values in one-qubit Hilbert spaces of a multiqubit quantum system. We construct many unextendible product bases (UPBs) of new sizes in such systems and provide a new construction of UPBs of $n$ qubits of cardinality $n+1$ when $n\equiv 3 \pmod{4}$. We also give a new method of constructing multiqubit entangled states with all partial transposes positive.