Some Aspects of Weighing Designs

Abstract

In a previous paper [8] the author proved that the $P_N$ and $S_N$ matrices are the most efficient weighing designs obtainable under Kishen's definition of efficiency [5], when $N$ is odd and $N \equiv 2 (\operatorname{mod} 4)$ respectively, subject to the conditions (i) The variances of the estimated weights are equal; (ii) The estimated weights are equally correlated. In this paper, assuming the above conditions, it is proved that the $P_N$ matrices are the best weighing designs under the definitions of Mood [6] and Ehrenfeld [2] when $N$ is odd, while the $S_N$ matrices are the best weighing designs under the definition of Ehrenfeld when $N \equiv 2 (\operatorname{mod} 4)$. Under Mood's definition of efficiency, the best weighing design $X,$ when $N \equiv 2 (\operatorname{mod} 4),$ is shown to be that for which $X'X = (N - 2)I_N + 2E_{NN},$ where $I_N$ is the $N$th order identity matrix and $E_{NN}$ is the $N$th order square matrix with positive unit elements everywhere. By applying the Hasse-Minkowski invariant, a necessary condition for the existence of the $S_N$ matrices is obtained, and the impossibilities of the $S_N$ matrices of orders 22, 34, 58 and 78 are shown.