  We consider two theorems from the theory of compressive sensing. Mainly a
theorem concerning uniform recovery of random sampling matrices, where the
number of samples needed in order to recover an $s$-sparse signal from linear
measurements (with high probability) is known to be $m\gtrsim s(\ln s)^3\ln N$.
We present new and improved constants together with what we consider to be a
more explicit proof. A proof that also allows for a slightly larger class of
$m\times N$-matrices, by considering what we call \emph{low entropy}. We also
present an improved condition on the so-called restricted isometry constants,
$\delta_s$, ensuring sparse recovery via $\ell^1$-minimization. We show that
$\delta_{2s}<4/\sqrt{41}$ is sufficient and that this can be improved further
to almost allow for a sufficient condition of the type $\delta_{2s}<2/3$.
