  We adapt Luk's analysis of the characteristic initial value problem in
General Relativity to the asymptotic characteristic problem for the conformal
Einstein field equations to demonstrate the local existence of solutions in a
neighbourhood of the set on which the data are given. In particular, we obtain
existence of solutions along a narrow rectangle along null infinity which, in
turn, corresponds to an infinite domain in the asymptotic region of the
physical spacetime. This result generalises work by K\'ann\'ar on the local
existence of solutions to the characteristic initial value problem by means of
Rendall's reduction strategy. In analysing the conformal Einstein equations we
make use of the Newman-Penrose formalism and a gauge due to J. Stewart.
