  Decay modes of excited nuclei are investigated in $^{78,82}$Kr + $^{40}$Ca
reactions at 5.5 MeV/nucleon. Charged products were measured by means of the
$4\pi$ INDRA array. Kinetic-energy spectra and angular distributions of
fragments with atomic number 3 $\le Z \le$ 28 indicate a high degree of
relaxation and are compatible with a fission-like phenomenon. Persistence of
structure effects is evidenced from elemental cross-sections ($\sigma_{Z}$) as
well as a strong odd-even-staggering (o-e-s) of the light-fragment yields. The
magnitude of the staggering does not significantly depend on the neutron
content of the emitting system. Fragment-particle coincidences suggest that the
light partners in very asymmetric fission are emitted either cold or at
excitation energies below the particle emission thresholds. The evaporation
residue cross-section of the $^{78}$Kr + $^{40}$Ca reaction is slightly higher
than the one measured in $^{82}$Kr + $^{40}$Ca reaction. The fission-like
component is larger by $\sim$ 25% for the reaction having the lowest
neutron-to-proton ratio. These experimental features are confronted to the
predictions of theoretical models. The Hauser-Feshbach approach including the
emission of fragments up to $Z$ = 14 in their ground states as well as excited
states does not account for the main features of $\sigma_{Z}$. For both
reactions, the transition-state formalism reasonably reproduces the
$Z$-distribution of the fragments with charge 12 $\le Z \le$ 28. However, this
model strongly overestimates the light-fragment cross-sections and does not
explain the o-e-s of the yields for 6 $\le Z \le$ 10. The shape of the whole
$Z$-distribution and the o-e-s of the light-fragment yields are satisfactorily
reproduced within the dinuclear system framework which treats the competition
between evaporation, fusion-fission and quasifission processes. The model
suggests that heavy fragments come mainly from quasifission while light
fragments are predominantly populated by fusion. An underestimation of the
cross sections for 16 $\le Z \le$ 22 could signal a mechanism in addition to
the capture process.
