20 04 the Maximal { − 1 , 1 } - Determinant of Order 15

We study the question of finding the maximal determinant of matrices of odd order with entries ±1. The most general upper bound on the maximal determinant, due to Barba, can only be achieved when the order is the sum of two consecutive squares. It is conjectured that the bound is always attained in such cases. Apart from these, only in orders 3, 7, 9, 11, 17 and 21 has the maximal value been established. In this paper we confirm the results for these orders, and add order 15 to the list. We follow previous authors in exhaustively searching for candidate Gram matrices having determinant greater than or equal to the square of a known lower bound on the maximum. We then attempt to decompose each candidate as the product of a {−1, 1}-matrix and its transpose. For order 15 we find four candidates, all of Ehlich block form, two having determinant (105 ·35 ·214)2 and the others determinant (108 ·35 ·214)2. One of the former decomposes (in an essentially unique way) while the remaining three do not. This result proves a conjecture made independently by W. D. Smith and J. H. E. Cohn. We also use our method to compute improved upper bounds on the maximal determinant in orders 29, 33, and 37, and to establish the range of the determinant function of {−1, 1}-matrices in orders 9 and 11.