  In this paper we consider random linear under-determined systems with
block-sparse solutions. A standard subvariant of such systems, namely,
precisely the same type of systems without additional block structuring
requirement, gained a lot of popularity over the last decade. This is of course
in first place due to the success in mathematical characterization of an
$\ell_1$ optimization technique typically used for solving such systems,
initially achieved in \cite{CRT,DOnoho06CS} and later on perfected in
\cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}. The success
that we achieved in \cite{StojnicCSetam09,StojnicUpper10} characterizing the
standard sparse solutions systems, we were then able to replicate in a sequence
of papers
\cite{StojnicCSetamBlock09,StojnicUpperBlock10,StojnicICASSP09block,StojnicJSTSP09}
where instead of the standard $\ell_1$ optimization we utilized its an
$\ell_2/\ell_1$ variant as a better fit for systems with block-sparse
solutions. All of these results finally settled the so-called threshold/phase
transitions phenomena (which naturally assume the asymptotic/large dimensional
scenario). Here, in addition to a few novel asymptotic considerations, we also
try to raise the level a bit, step a bit away from the asymptotics, and
consider the finite dimensions scenarios as well.
