A unified approach in addition or deletion of two level factorial designs

Suppose it is desired to have an optimal resolution III fraction of a 2p factorial in n runs where n[reverse not equivalent]1 (mod 4) or n[reverse not equivalent]3 (mod 4). If n[reverse not equivalent]1 (mod 4), we have to decide if we should add a run in a nxp submatrix of a Hadamard matrix of order n, say Hn or, alternatively, if we should delete three runs from a (n+4)xp submatrix of a Hadamard matrix of order n+4, say Hn+4, in an optimal manner, respectively. Similarly, when n[reverse not equivalent]3 (mod 4), we have to decide between optimally adding three more runs to a nxp submatrix of Hn or optimally deleting a single run from a (n+4)xp submatrix of Hn+4. The question to be studied is whether both strategies give designs that are equally efficient in terms of a well defined optimality criterion. We show that, in both cases, for p=3 both strategies give equally efficient designs under the D- or the A-optimality criterion. When n[reverse not equivalent]1 (mod 4) and p>3, both criteria show that the "addition" design is always better than the "deletion" design. However, when n[reverse not equivalent]3 (mod 4) and p>3, the choice of the most efficient design varies as p enlarges.