  In this paper, we study the restricted isometry property of partial random
circulant matrices. For a bounded subgaussian generator with independent
entries, we prove that the partial random circulant matrices satisfy $s$-order
RIP with high probability if one chooses $m\gtrsim s \log^2(s)\log (n)$ rows
randomly where $n$ is the vector length. This improves the previously known
bound $m \gtrsim s \log^2 s\log^2 n$.
