Balanced incomplete block designs (BIBD) show each item (a) equally often, (b) equally often with each other item and (c) equally often in each position.
Standard published tables of BIBDs exist.
However, there is not necessarily a perfect BIBD for each total number of items or for each total number of items shown a specific number of times in choice sets of specific sizes. For this reason, experimental design software (like SAS, Ngene and our Lighthouse Studio software) allow a user to do computerized searches for near-perfect BIBDs for any number of items shown any number of times each.
Of course if a BIBD does exist for a specific experiment, it is very easy in our software for a user to import it and use it instead of the design our software would make.