Inseparable orthogonal matrices over \(\mathbb{Z}_2\) (Q1273664)

A \((0,1)\)-matrix \(A\) is called orthogonal over \(\mathbb{Z}_2\) if both \(AA^T\) and \(A^TA\) are diagonal matrices. A marix \(A\) is called inseparable if \(A\) contains no zero row or zero column and there do not exist permutation matrices \(P\) and \(Q\) such that \[ PAQ= \begin{pmatrix} B&0\\ 0&C\end{pmatrix}. \] A matrix \(A\) is said to be of type 0 if \(AA^T= 0\) and \(A^TA=0\). A square matrix \(A\) of order \(n\) is said to be of type 1 if \(AA^T=I_n\). It turns out that an inseparable orthogonal matrix over \(\mathbb{Z}_2\) is either of type 0 or of type 1. Let \(f_0(m,n)\) (respectively, \(F_0(m,n)\)) denote the smallest (respectively, largest) number of 1's in an \(m\times n\) inseparable orthogonal matrix of type 0 over \(\mathbb{Z}_2\), and \(f_1(n)\) (respectively, \(F_1(n)\)) denote the smallest (respectively largest) number of 1's in an \(n\times n\) inseparable orthogonal matrix of type 1 over \(\mathbb{Z}_2\). The formulas for \(f_0(m,n)\), \(F_0(m,n)\), \(f_1(n)\), and \(F_1(n)\) are completely determined.