The maximum determinant of 21\(\times 21\) \((+1,-1)\)-matrices and D-optimal designs (Q579811)

In this paper the authors give a square matrix \(R^*\) of size 21 with elements -1, \(+1\) such that det \(M^*=\det (R^{*T}R^*)=20^{18}(116)^ 2\), and that \(R^*\) is the D-optimal design, i.e. the matrix \(R^*\) has the maximum determinant. In order to prove this D-optimality, a computational procedure is given to obtain all (21\(\times 21)\) symmetric and positive-definite matrices \(M=(m_{ij})\), \(m_{ii}=21\), \(m_{ij}\equiv 1(mod 4)\), \(i\neq j\), such that det (M)\(=square\) of an integer and det \(M\geq 20^{18}(116)^ 2\). It is shown that, besides \(M^*\), there are two more such matrixes \(M_ 1\), \(M_ 2\) which are shown not to exist.