  The antenna subtraction method has achieved remarkable success in various
processes relevant to the Large Hadron Collider. In Reference [1], an algorithm
was proposed for constructing real-radiation antenna functions for
electron-positron annihilation, directly from specified unresolved limits,
accommodating any number of real emissions. Here, we extend this algorithm to
build antennae involving partons in the initial state, specifically the
initial-final and initial-initial antennae. Using this extended algorithm, we
explicitly construct all NLO QCD antenna functions and compare them with
previously extracted antenna functions derived from matrix elements.
Additionally, we rigorously match the integration of the antenna functions over
the initial-final and initial-initial unresolved phase space with the previous
approach, providing an independent validation of our results. The improved
antenna functions are more compact and reduced in number, making them more
readily applicable for higher-order calculations.
