Unextendible product bases correspond to unextendible orthogonal matrices (UOMs), which are a key notion in quantum information theory. We investigate the problem of constructing \(13\times 9\) UOMs \(U_{13,9}\) . We present the notions of full, maximum and minimum columns to characterize a general \(U_{13,9}\) . We show that every column of \(U_{13,9}\) has at least two pairs of orthogonal variables, and \(U_{13,9}\) does not have nine full columns. We also show that the multiplicity of each variable in a column of \(U_{13,9}\) is at most four. When this upper bound is saturated for four variables, it turns out that they respectively have multiplicities (4, 4, 4, 1), (4, 4, 3, 2), (4, 3, 4, 2), or (4, 3, 3, 3). To exclude the existence of \(U_{13,9}\) with any one of them, we present the notion of diagonal-type matrices with four types. None of these types turn out to be a submatrix of \(U_{13,9}\) . They imply that no \(U_{13,9}\) has at the same time eight full columns. We further show that every column of \(U_{13,9}\) contains at least three and at most six independent variables. Our results stand for the latest progress on UOMs and help construct \(U_{13,9}\) by programs.
