On the existence of flat orthogonal matrices

Abstract

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem.

10 pages