  We describe algorithms for computing maximal determinants of binary circulant
matrices of small orders. Here "binary matrix" means a matrix whose elements
are drawn from $\{0,1\}$ or $\{-1,1\}$. We describe efficient parallel
algorithms for the search, using Duval's algorithm for generation of necklaces
and the well-known representation of the determinant of a circulant in terms of
roots of unity. Tables of maximal determinants are given for orders $\le 53$.
Our computations extend earlier results and disprove two plausible conjectures.
