Frustratingly, attempts to discretize space invariably run into problems with relativity, since they effectively impose a preferred frame of reference. I.e. you can impose a minimum distance, but relativistic length contraction means that observers measure different minima and in different directions.

Apparently, under some of these models, this implies that the speed of light ends up depending on wavelength, lending them to empirical tests. My understanding is that these discrete space models have failed to line up with experiment, at least within the limits of measurement.