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D-optimal weighing designs for four and five objects. (English) Zbl 0930.62076

Let \(M_{j,d} (0,1)\) be the set of all \(j\times d\;(0,1)\) real matrices. For \(j=4,5\), and all \(d\geq j\), the authors determine the value of \(G(j,d)= \max\{ \text{det} AA^T: A\in M_{j,d} (0,1)\}\), and find matrices for which the maximum is attained. This maximization problem arises in at least two contexts – finding \(D\)-optimal weighing designs for weighing four and five objects, and maximizing the volume of a \(j\)-simplex in the \(d\)-dimensional cube \([0,1]^d\). The results of this paper establish that lower bounds for \(G(j,d)\) derived by M. Hudelson, V. Klee, and D. Larman [HKL] [Linear Algebra Appl. 241-243, 519-598 (1996; Zbl 0861.15004)] are in fact equal to \(G(j,d)\) for \(j=4,5\). The proof proceeds by first defining \[ S_{j,d} =\begin{cases} \bigl\{A\in M_{j,d} (0,1): e_jA= ke_d\bigr\} \quad & \text{if } j=2k-1\\ \bigl\{ A\in M_{j,d} (0,1): e_jA= ke_d\text{ or }(k+1) e_d\bigr\} \quad & \text{if }j= 3k, \end{cases} \] where \(e_n\) is the vector of all-ones of length \(n\), and \[ F(j, d)=\max \bigl\{\text{det} A A^T\mid A\in S_{j,d} (0,1)\bigr\}. \] It is then established that, for \(j=4\) and all \(d\geq 4\), every \(D\)-optimal matrix in \(M_{j, d}(0,1)\) is actually in \(S_{j,d} (0,1)\). Next, it is shown that \(G(5,d)= F(5,d)\) for all \(d\geq 5\), except perhaps for \(d=5\), 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 33, 34, 35, 36, 37, 44, 45, 46. Finally, these exceptional cases are analyzed and it is proved that, in every case considered, the lower bounds for \(G(5,d)\) given in [HKL] are in fact equal to \(G(5,d)\). In the process the authors also find matrices \(A\) that attain the values of \(G(j,d)\) for \(j=4,5\) and all \(d\geq j\).

MSC:

62K05 Optimal statistical designs
05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
05B05 Combinatorial aspects of block designs
15A15 Determinants, permanents, traces, other special matrix functions

Citations:

Zbl 0861.15004
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