The D-optimal saturated designs of order 22

This paper attempts to prove the D-optimality of the saturated designs X∗ and X∗∗ of order 22, already existing in the current literature. The corresponding non-equivalent information matrices M∗=(X∗)TX∗ and M∗∗=(X∗∗)TX∗∗ have the maximum determinant. Within the application of a specific procedure, all symmetric and positive definite matrices M of order 22 with determinant the square of an integer and ≥det(M∗) are constructed. This procedure has indicated that there are 26 such non-equivalent matrices M, for 24 of which the non-existence of designs X such that XTX =M is proved. The remaining two matrices M are the information matrices M∗ and M∗∗.