  A unitary t-design is a set of unitaries that is "evenly distributed" in the
sense that the average of any t-th order polynomial over the design equals the
average over the entire unitary group. In various fields -- e.g. quantum
information theory -- one frequently encounters constructions that rely on
matrices drawn uniformly at random from the unitary group. Often, it suffices
to sample these matrices from a unitary t-design, for sufficiently high t. This
results in more explicit, derandomized constructions. The most prominent
unitary t-design considered in quantum information is the multi-qubit Clifford
group. It is known to be a unitary 3-design, but, unfortunately, not a
4-design. Here, we give a simple, explicit characterization of the way in which
the Clifford group fails to constitute a 4-design. Our results show that for
various applications in quantum information theory and in the theory of convex
signal recovery, Clifford orbits perform almost as well as those of true
4-designs. Technically, it turns out that in a precise sense, the 4th tensor
power of the Clifford group affords only one more invariant subspace than the
4th tensor power of the unitary group. That additional subspace is a stabilizer
code -- a structure extensively studied in the field of quantum error
correction codes. The action of the Clifford group on this stabilizer code can
be decomposed explicitly into previously known irreps of the discrete
symplectic group. We give various constructions of exact complex projective
4-designs or approximate 4-designs of arbitrarily high precision from Clifford
orbits. Building on results from coding theory, we give strong evidence
suggesting that these orbits actually constitute complex projective 5-designs.
