  In our recent work \cite{StojnicCSetam09} we considered solving
under-determined systems of linear equations with sparse solutions. In a large
dimensional and statistical context we proved that if the number of equations
in the system is proportional to the length of the unknown vector then there is
a sparsity (number of non-zero elements of the unknown vector) also
proportional to the length of the unknown vector such that a polynomial
$\ell_1$-optimization technique succeeds in solving the system. We provided
lower bounds on the proportionality constants that are in a solid numerical
agreement with what one can observe through numerical experiments. Here we
create a mechanism that can be used to derive the upper bounds on the
proportionality constants. Moreover, the upper bounds obtained through such a
mechanism match the lower bounds from \cite{StojnicCSetam09} and ultimately
make the latter ones optimal.
