A computer study was made of the spillover, crosspolarisation and illumination efficiency of paraboloids of various focal-length/paraboloid-diameter (F/D) ratios, fed by rectangular, square and circular wave-guides in their lowest-order modes. The feed radiation patterns used were those derived by simple diffraction theory from the incident waveguide-mode field distribution. The approximations made in this derivation give unnaturally low spillovers for very small aperture feeds, and thus favour shortF/Dsystems, which would use such feeds. The feed dimensions were optimised to give a maximum figure of merit (gain/total noise temperature) for each combination of feed, receiver noise temperature andF/Dratio. Some of the optimum horn dimensions and resulting illumination tapers, excess aerial noise temperatures, gain efficiencies and figures of merit produced by the computations are presented. The results show, among other things, that the figure of merit of aF/D-ratio system. about 0.8dB less than that of a large-F/D-ratio system They also show that, if the system has a low receiver temperature, the feed dimensions should be larger than conventional design procedures indicate, to give up to 3dB more edge-illumination taper