On optimal third order rotatable designs

We obtain results for choosing optimal third order rotatable designs for the fitting of a third order polynomial response surface model, for m greater than or equal to 3 factors. By representing the surface in terms of Kronecker algebra, it can be established that the two parameter family of boundary nucleus designs forms a complete class, under the Loewner matrix ordering. In this paper, we first narrow the class further to a smaller complete class, under the componentwise eigenvalue ordering. We then calculate specific optimal designs under Kiefer's phi(p)-criteria (which include the often used E, A-, and D-criteria). The E-optimal design attains a particularly simple, explicit form.