  We discuss recent progress in the calculation of integrated antenna functions
for final-state radiation at N$^3$LO in QCD. Antenna functions are directly
extracted from physical matrix elements for the decay of a colour-singlet state
into partons. In order to compute their integrated counterparts, which are
necessary ingredients for fixed-order calculations in QCD, each allowed final
state has to be individually integrated over the inclusive phase space of the
unresolved radiation. We analytically integrated two-, three-, four- and
five-particle final-state matrix elements for the decay of a virtual photon and
a Higgs boson into partons, considering up to third-order corrections in the
strong coupling constant $\alpha_s$. The integration over the inclusive phase
space is performed relying on well-established reverse unitarity and IBP
reduction techniques. We conclude with preliminary comments about the structure
of N$^3$LO subtraction terms in the context of the antenna subtraction method.
