  We study spin 3/2 fermionic cold atoms with attractive interactions confined
in a one-dimensional optical lattice. Using numerical techniques, we determine
the phase diagram for a generic density. For the chosen parameters,
one-particle excitations are gapped and the phase diagram is separated into two
regions: one where the two-particle excitation gap is zero, and one where it is
finite. In the first region, the two-body pairing fluctuations (BCS) compete
with the density ones. In the other one, a molecular superfluid (MS) phase, in
which bound-states of four particles form, competes with the density
fluctuations. The properties of the transition line between these two regions
is studied through the behavior of the entanglement entropy. The physical
features of the various phases, comprising leading correlations, Friedel
oscillations, and excitation spectra, are presented. To make the connection
with experiments, the effect of a harmonic trap is taken into account. In
particular, we emphasize the conditions under which the appealing MS phase can
be realized, and how the phases could be probed by using the density profiles
and the associated structure factor. Lastly, the consequences on the flux
quantization of the different nature of the pairing in the BCS and MS phases
are studied in a situation where the condensate is in a ring geometry.
