  We present a lattice QCD spectroscopy study in the isospin singlet,
strangeness $-2$ sectors relevant for the conjectured $H$ dibaryon. We employ
both local and bilocal interpolating operators to isolate the ground state in
the rest frame and in moving frames. Calculations are performed using two
flavors of O($a$)-improved Wilson fermions and a quenched strange quark. Our
initial point-source method for constructing correlators does not allow for
bilocal operators at the source; nevertheless, results from using these
operators at the sink indicate that they provide an improved overlap onto the
ground state in comparison with the local operators. We also present results,
in the rest frame, using a second method based on distillation to compute a
hermitian matrix of correlators with bilocal operators at both the source and
the sink. This method yields a much more precise and reliable determination of
the ground-state energy. In the flavor-SU(3) symmetric case, we apply
L\"uscher's finite-volume quantization condition to the rest-frame and
moving-frame energy levels to determine the $S$-wave scattering phase shift,
near and below the two-particle threshold. For a pion mass of 960 MeV, we find
that there exists a bound $H$ dibaryon with binding energy
${\Delta}E=(19\pm10)$ MeV. In the 27-plet (dineutron) sector, the finite-volume
analysis suggests that the existence of a bound state is unlikely.
