Genesis IIIs were a line of supercomputers.
A P-series Genesis III was in operation on board Space Station Alpha. It was tasked with finding new places for humanity to colonise. It formulated a way to do this by miniaturising the humans with a Composition Adjustment Teleport, and having them live on a single atom of carbon.
The Genesis III brought the Fourth Doctor to Space Station Alpha in the hope and using him and his TARDIS to continue this miniaturisation. The Genesis III agreed to be linked to the Doctor in order to gain his knowledge. However, the Doctor told him the story of Puss in Boots, confusing the computer and rendering it harmless. The Doctor explained that this was due to it not being able to imagine; this would not be incorporated until the R-series. (PROSE: Inter-Galactic Cat)