  We give a short argument that yields a new lower bound on the number of
subsampled rows from a bounded, orthonormal matrix necessary to form a matrix
with the restricted isometry property. We show that a matrix formed by
uniformly subsampling rows of an $N \times N$ Walsh matrix contains a
$K$-sparse vector in the kernel, unless the number of subsampled rows is
$\Omega(K \log K \log (N/K))$ -- our lower bound applies whenever $\min(K, N/K)
> \log^C N$. Containing a sparse vector in the kernel precludes not only the
restricted isometry property, but more generally the application of those
matrices for uniform sparse recovery.
