Values of Minors of an Infinite Family of D-Optimal Designs and Their Application to the Growth Problem

We obtain explicit formulae for the values of the 2v — j minors, j = 0, 1, 2 of D-optimal designs of order 2v = x2 + y2, v odd, where the design is constructed using two circulant or type 1 incidence matrices of either two SBIBD(2s2 + 2s + 1, s2, s2-s/2) or 2 — {2s2 + 2s + 1; s2, s2; s(s–1)} sds. This allows us to obtain information on the growth problem for families of matrices with moderate growth. Some of our theoretical formulae imply growth greater than 2(2s2 + 2s + 1) but experimentation has not yet supported this result. An open problem remains to establish whether the (1, -1) CP incidence matrices of certain SBIBDs and 2 — {2s2 + 2s + 1; s2, s2; s(s — 1)} sds which yield D—optimal designs, can have growth greater than 2v.