On the existence of flat orthogonal matrices

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