  I review some aspects of antenna subtraction at next-to-next-to-leading order
(NNLO) in QCD and provide motivation for its extension to N$^3$LO. Next, I
introduce the antenna functions required for the construction of infrared
counterterms for final-state radiation at this order. Lastly, I describe the
evaluation of the antenna functions and their phase-space integration, first
presented in [1-3], and I elaborate on their application to precision
observables in jet production at lepton colliders.
