N=40 det=6.439842994660582748450 e31 circ=-+++---+-++---++-+-+---------++-++--+-++ think I'll quit finding them now. ---- In summary, all 2^N different NxN circulant sign matrices can have their determinants computed in O(N) arithmetic operations per matrix using the fourier trick combined with the gray code trick. If however we were to generate "binary necklaces" via a "3-gray code" for them, Mark Weston & Vincent Vajnovszki: Gray codes for necklaces and Lyndon words of arbitrary base, Pure Mathematics and Applications 17, 1-2 (2006) 175-182. http://homelinux.capitano.unisi.it/~puma/17_1_2/weston.pdf T.Ueda: Gray codes for necklaces, Discrete Maths. 219,1-3 (2000) 235-248. T.M.Y. Wang and C.D. Savage: A Gray code for necklaces of fixed density, SIAM J. Discrete Maths 9,4 (1996) 654-673. then we could effectively compute the determinants of all NxN circulant sign matrices in O(1) arithmetic operations per matrix, which would have enabled me to reach, not N=40, but in fact about N=44, with the same amount of computing. Perhaps somebody will program that refinement.