  Our companion work \cite{Stojnicl1BnBxasymldp} considers random
under-determined linear systems with box-constrained sparse solutions and
provides an asymptotic analysis of a couple of modified $\ell_1$ heuristics
adjusted to handle such systems (we refer to these modifications of the
standard $\ell_1$ as binary and box $\ell_1$). Our earlier work
\cite{StojnicISIT2010binary} established that the binary $\ell_1$ does exhibit
the so-called phase-transition phenomenon (basically the same phenomenon
well-known through earlier considerations to be a key feature of the standard
$\ell_1$, see, e.g.
\cite{DonohoPol,DonohoUnsigned,StojnicCSetam09,StojnicUpper10}). Moreover, in
\cite{StojnicISIT2010binary}, we determined the precise location of the
co-called phase-transition (PT) curve. On the other hand, in
\cite{Stojnicl1BnBxasymldp} we provide a much deeper understanding of the PTs
and do so through a large deviations principles (LDP) type of analysis. In this
paper we complement the results of \cite{Stojnicl1BnBxasymldp} by leaving the
asymptotic regime naturally assumed in the PT and LDP considerations aside and
instead working in a finite dimensional setting. Along the same lines, we
provide for both, the binary and the box $\ell_1$, precise finite dimensional
analyses and essentially determine their ultimate statistical performance
characterizations. On top of that, we explain how the results created here can
be utilized in the asymptotic setting, considered in
\cite{Stojnicl1BnBxasymldp}, as well. Finally, for the completeness, we also
present a collection of results obtained through numerical simulations and
observe that they are in a massive agreement with our theoretical calculations.
