Computation of Maximal Determinants of Binary Circulant Matrices

TL;DR

This paper presents efficient algorithms for computing the maximal determinants of small binary circulant matrices, providing new tables of results that extend previous findings and challenge existing conjectures.

Contribution

It introduces parallel algorithms utilizing necklace generation and roots of unity to compute maximal determinants, extending known results and disproving some conjectures.

Findings

Tables of maximal determinants for matrices of order up to 53.

Disproof of two plausible conjectures about binary circulant matrices.

Extension of previous computational results in the field.

Abstract

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.