Constant used to determine the minimum samples in Compressed Sensing
The constant C is usually left unspecified as it can be very hard to calculate.  It is independent of the signal dimension, and results in Compressed Sensing are generally only concerned with big-O results, so it falls out when we say something like $m = \mathcal{O}(s \ln{(N/s)})$.  The important part in this statement is the strong dependence on the sparsity level, and that the signal dimension $N$ only logarithmically affects the number of measurements. 

To give an example of where such a constant would come from:  Let $V_d$ be the volume of the unit $\ell_2$ ball in $\mathbb{R}^d$.  Some simple cases are that $V_2 = \pi r^2$ and $V_3 = \frac{4}{3} \pi r^3$.  Note that both of these results are of the form $V_d = C r^d$, and this pattern generalizes for finite $d$.  In CS, it is $r^d$ that we are interested in, as it captures the dependence on the signal dimension.  

So $C$ will vary depending on the algorithm as different inequalities may be used in the proofs.  I don't know of any publications that specifically calculate C.