  We have extended our previous computations of the even-N moments of the
flavour-singlet four-loop splitting functions to N = 12 for the pure-singlet
quark case and N = 10 for all other cases. These results, obtained using
physical quantities in inclusive deep-inelastic scattering, have been and will
be used to validate conceptionally much more challenging determinations of
these splitting functions from off-shell operator matrix elements (OMEs). For
the quark-gluon and gluon-gluon splitting functions, which have yet to be
computed to higher N using OMEs, we construct approximations based on our
moments and endpoint constraints, where we present new large-x results for the
gluon-gluon case. These approximations facilitate an approximate N^3LO
evolution of parton distributions which are sufficiently accurate outside the
region of small momentum fractions x.
