D-optimal weighing designs for n≡−1mod4 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)=maxdetXTX:X∈Mm,n(0,1). Inthis paper we exhibit some new formulas for G(m,n) where n≡−1(mod4). 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.