Tools for constructing optimal two-level factorial designs for a linear model containing main effects and one two-factor interaction

In Hedayat and Pesotan [1992, Two-level factorial designs for main effects and selected two-factor interactions. Statist. Sinica 2, 453-464.] the concepts of a g(n,e)-design and a g(n,e)-matrix are introduced to study designs of n factor two-level experiments which can unbiasedly estimate the mean, the n main effects and e specified two-factor interactions appearing in an orthogonal polynomial model and it is observed that the construction of a g-design is equivalent to the construction of a g-matrix. This paper deals with the construction of D-optimal g(n,1)-matrices. A standard form for a g(n,1)-matrix is introduced and some lower and upper bounds on the absolute determinant value of a D-optimal g(n,1)-matrix in the class of all g(n,1)-matrices are obtained and an approach to construct D-optimal g(n,1)-matrices is given for 2⩽n⩽8. For two specific subclasses, namely a certain class of g(n,1)-matrices within the class of g(n,1)-matrices of index one and the class C(H) of g(8t+2,1)-matrices constructed from a normalized Hadamard matrix H of order 8t+4(t⩾1) two techniques for the construction of the restricted D-optimal matrices are given.