  We classify the orientable finite-volume hyperbolic 3-manifolds having
non-empty compact totally geodesic boundary and admitting an ideal
triangulation with at most four tetrahedra. We also compute the volume of all
such manifolds, we describe their canonical Kojima decomposition, and we
discuss manifolds having cusps.
  The manifolds built from one or two tetrahedra were previously known. There
are 151 different manifolds built from three tetrahedra, realizing 18 different
volumes. Their Kojima decomposition always consists of tetrahedra (but
occasionally requires four of them). And there is a single cusped manifold,
that we can show to be a knot complement in a genus-2 handlebody. Concerning
manifolds built from four tetrahedra, we show that there are 5033 different
ones, with 262 different volumes. The Kojima decomposition consists either of
tetrahedra (as many as eight of them in some cases), or of two pyramids, or of
a single octahedron. There are 30 manifolds having a single cusp, and one
having two cusps.
  Our results were obtained with the aid of a computer. The complete list of
manifolds (in SnapPea format) and full details on their invariants are
available on the world wide web.
