Almost D-optimal designs

Abstract

A D-optimal design of order n is an n × n matrix with entries ±1 having maximal determinant, g(n). Although good upper bounds for g(n) are known, it is not always easy to determine g(n), much less to find a suitable design. The term "almost D-optimal" has recently been used to refer to a design whose determinant is close to the known bounds, perhaps with the implicit conjecture that it is actually D-optimal. We consider the cases n = 22, 15 and 19 in this sense, and show why when n ≡ 3 ( mod 4 ) the best known upper bound cannot usually be attained.