ABSTRACT
An m × n matrix W whose entries are all 1 or −1 is called a (±1)-design matrix; if the entries of W are 0 or 1, then W is a (0, 1)-design matrix. Each design matrix corresponds to a weighing design. That is, a scheme for estimating the weights of n objects in m weighings. Since the weights of n objects cannot be estimated in fewer than n weighings, we consider only those pairs (m, n) with m ≥ n. The rows of W encode a two-pan or one-pan weighing design with n objects x 1, …, xn being weighed in m weighings. If W ∈ {±1} m×n , an entry of 1 in the (i, j)-th position of W indicates that object xj is put in the right pan in the i-th weighing while an entry of −1 means that xj is placed in the left pan. If W ∈ {0, 1} m×n , an entry of 1 in the (i, j)-th position indicates that object xj is included in the i-th weighing while an entry of 0 means that the object is not included. In the presence of errors for the scale, we can expect only to find estimators ŵ 1, …, ŵn for the actual weights w 1, …, wn of the objects. We want to choose a weighing design that is optimal with respect to some condition, an idea going back to Hotelling [Hot44] and Mood [Moo46]. See also [HS79] and [Slo79]. Under certain assumptions on the error of the scale, we can express optimality conditions in terms of WT W (see [Puk93]). The value of det WT W is inversely proportional to the volume of the confidence region of the estimators of the weights of the objects. Thus, matrices for which det WT W is large correspond to weighing designs that are desirable.
