D - optimal weighing designs for n ≡ − 1 ( mod 4 ) objects and a large number of weighings

Let Mm,n(0, 1) denote the set of all m× n (0,1)-matrices and let G(m,n) = max{detXX : X ∈Mm,n(0, 1)}. In this paper we exhibit some new formulas for G(m,n) where n ≡ −1 (mod 4). Specifically, for m = nt+r where 0 ≤ r < n, we show that for all sufficiently large t, G(nt+r, n) is a polynomial in t of degree n that depends on the characteristic polynomial of the adjacency matrix of a certain regular graph. Thus the problem of finding G(nt+ r, n) for large t is equivalent to finding a regular graph, whose degree of regularity and number of vertices depend only on n and r, with a certain “trace-minimal” property. In particular we determine the appropriate trace-minimal graph and hence the formulas for G(nt+ r, n) for n = 11, 15, all r, and all sufficiently large t.