  We extend the techniques of H\"{u}gel, Rauhut and Strohmer (Found. Comput.
Math., 2014) to show that for every $\delta\in(0,1]$, there exists an explicit
random $m\times N$ partial Fourier matrix $A$ with
$m=s\operatorname{polylog}(N/\epsilon)$ and entropy
$s^\delta\operatorname{polylog}(N/\epsilon)$ such that for every $s$-sparse
signal $x\in\mathbb{C}^N$, there exists an event of probability at least
$1-\epsilon$ over which $x$ is the unique minimizer of $\|z\|_1$ subject to
$Az=Ax$. The bulk of our analysis uses tools from decoupling to estimate the
extreme singular values of the submatrix of $A$ whose columns correspond to the
support of $x$.
