Journal of Integer Sequences, Vol. 21 (2018), Article 18.5.6

Computation of Maximal Determinants of Binary Circulant Matrices

Richard P. Brent
Mathematical Sciences Institute
Australian National University
Canberra, ACT 2601

Adam B. Yedidia
Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Cambridge, MA 02139


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 ≤ 52. Our computations extend earlier results and disprove two plausible conjectures.

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(Concerned with sequences A000031 A086432 A215723 A215897.)

Received April 25 2018; revised version received June 6 2018. Published in Journal of Integer Sequences, June 10 2018.

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