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We have computed the even-$N$ moments $N \leq 20$ of the gluon-to-quarksplitting function $P_{\rm qg}$ at the fourth order of perturbative QCD via therenormalization of off-shell operator matrix elements. Our results, derivedanalytically for a general gauge group, agree with all results obtained forthis function so far, in particular with the lowest five moments obtained viaphysical cross sections. Using our new moments and all available endpointconstraints, we construct approximations for the four-loop $P_{\rm qg}(x)$ thatshould be sufficient for a wide range of collider-physics applications. TheN$^3$LO corrections resulting from these and the corresponding quark-quarksplitting functions lead to a marked improvement of the perturbative accuracyfor the scale derivative of the singlet quark distribution, with effects of 1%or less at $x \gtrsim 10^{\,-4}$ at a standard reference scale with $\alpha_s =0.2$.

Four-loop splitting functions in QCD -- The gluon-to-quark case