Determinants of matrices of the conference type

An n by n conference type matrix has 0ʹs on the main diagonal and ±1ʹs elsewhere. We investigate the largest possible determinant of such a matrix. The literature is extensive for n even, but for n odd the question has not been previously studied. We determine the maxima up to n = 11: for n = 3, 5, 7, 9, 11 they are 2, 22, 394, 8760, 240786. It will require a new idea or a huge computation to go beyond n = 11.