1. The determinant of any NxN sign matrix is divisible by 2^(N-1). Proof: subtract 1st row from other rows, result is all rows except 1st are divisible by 2, hence by using expansion by minors proof follows. 2. The maximum |determinant| among NxN circulant sign matrices if N is sufficiently large, is at least (c*N)^(N/2) for some constant c>0 which I have not bothered to work out, but c=0.1 should be safe. Proof sketch: Use the FFT-product formula for determinant. Need to prove signs exist causing each of the N |multiplicands| to be
c*sqrt(N). This can be done using a counting argument and central limit theorem - if |c|>0 small enough then at least half the sign-choices survive the kth demand, and due to the unitary nature of the FFT matrix each demand is orthogonal to the previous, hence effects can be reckoned independently. Initially 2^N sign choices, and after N demands at least one remains.