The A-optimal non-saturated weighing designs for n = 15 observations

The problem of constructing A-optimal weighing and first order fractional factorial designs for n ≡ 3 mod 4 observations is considered. The non-existence of the weighing design matrices for n = 15 observations and k = 13, 14 factors, for which the corresponding information matrices have inverses with minimum trace, is proved. These designs are the first non-saturated cases (k < n) in which the unattainability of Sathe and Shenoy's (1989) lower bound on A-optimality is shown. Using an algorithm proposed in Farmakis (1991) we construct 15 × k (+1, −1)-matrices for k = 13, 14 and we prove their A-optimality using the improved (higher) lower bounds on A-optimality established by Chadjiconstantinidis and Kounias (1994). Also the A-optimal designs for n = 15, k ⩽ 12 are given.