General Lower Bounds on Maximal Determinants of Binary Matrices

Abstract

We give general lower bounds on the maximal determinant of $n \times n$ $\{+1,-1\}$-matrices, both with and without the assumption of the Hadamard conjecture. Our bounds improve on earlier results of de Launey and Levin (2010) and, for certain congruence classes of $n \bmod 4$, the results of Koukouvinos, Mitrouli and Seberry (2000). In an Appendix we give a new proof, using Jacobi's determinant identity, of a result of Szöllősi (2010) on minors of Hadamard matrices.