To my knowledge, there are very few concrete proposals for what a discrete universe would actually look like. Most physicists believe that the universe should be symmetric under lorentz transforms, which rules out all the obvious ways like splitting the universe into little cubes.
Although discrete models may not exist, discreetness is expected because without there's likely unsolvable informational problems.
This is why Beckenstein was so brilliant, he was among the first to understand physical systems must fit into information theory. If they don't, physics has far worse and profound issues.
Any description of the universe is informational. We should hope that the universe is informational so that humanity can have a complete description of the cosmos. If the universe is not informational, since description by nature is informational, there would be no way to describe the universe completely.
