  We present an implementation of the Bethe-Salpeter equation (BSE) for
core-conduction band pairs within the framework of the projector augmented-wave
method. For validation, the method is applied to the $K$-edges of diamond,
graphite, hexagonal boron-nitride, as well as four lithium-halides (LiF, LiCl,
LiI, LiBr). We compare our results with experiment, previous theoretical BSE
results, and the density functional theory-based supercell core-hole method. In
all considered cases, the agreement with experiment is excellent, in particular
for the position of the peaks as well as the fine structure. Comparing BSE to
supercell core-hole spectra we find that the latter often qualitatively
reproduces the experimental spectrum, however, it sometimes lacks important
details. This is shown for the $K$-edges of diamond and nitrogen in hexagonal
boron-nitride, where we are capable to resolve within the BSE experimental
features that are lacking in the core-hole method. Additionally, we show that
in certain systems the supercell core-hole method performs better if the
excited electron is added to the background charge. We attribute this improved
performance to a reduced self-interaction.
