A note on approximate Hadamard matrices

A Hadamard matrix is a scaled orthogonal matrix with +/- 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 whenn>4. Recently, Dong and Rudelson proved the existence of approximate Hadamard matrices in all dimensions: there exist universal 0<c<C= 1, there is a matrix A is an element of{-1,1}nxn satisfying, for allx is an element of Rn, c root n & Vert;x & Vert;2 <=& Vert;Ax & Vert;2 <= C root n & Vert;x & Vert;2. We observe that, as a consequence of the existence of flat Littlewood polynomials, circulant approximate Hadamard matrices exist for all n >= 1 .