I present an extension of the NNPDF4.0 parton distribution
functions (PDFs) at approximate next-to-next-to-next-to-leading order
(N3LO). In particular, I discuss how an approximation for the N3LO
splitting functions can be constructed from available fixed-order and
resummed computations, and how this can be combined with (partial)
knowledge of N3LO partonic matrix elements to determine a set of N3LO
PDFs. I explain how this set incorporates uncertainties from
incomplete knowledge of N3LO terms in the splitting functions,
and from missing higher-order corrections, estimated by scale
variation, through a theory covariance matrix formalism. I assess the
perturbative stability of the PDFs, compare them with those from MSHT,
and examine the phenomenological impact of approximate N3LO
corrections on parton luminosities at the LHC, and on the Higgs and
Drell-Yan total production cross sections.