A note on approximate Hadamard matrices

A Hadamard matrix is a scaled orthogonal matrix with \(\pm 1\) entries. Such matrices exist in certain dimensions: the Hadamard conjecture is that such a matrix always exists when n is a multiple of 4. A conjecture attributed to Ryser is that no circulant Hadamard matrices exist when \(n > 4\). Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal \(0< c< C < \infty \) so that for all \(n \ge 1\), there is a matrix \(A \in \left\{ -1,1\right\} ^{n \times n}\) satisfying, for all \(x \in \mathbb {R}^n\),

$$\begin{aligned} c \sqrt{n} \Vert x\Vert _2 \le \Vert Ax\Vert _2 \le C \sqrt{n} \Vert x\Vert _2. \end{aligned}$$

We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all \(n \ge 1\).