  Quantum contextuality supports quantum computation and communication. One of
its main vehicles is hypergraphs. The most elaborated are the Kochen-Specker
ones, but there is also another class of contextual sets that are not of this
kind. Their representation has been mostly operator-based and limited to
special constructs in three- to six-dim spaces, a notable example of which is
the Yu-Oh set. Previously, we showed that hypergraphs underlie all of them, and
in this paper, we give general methods - whose complexity does not scale up
with the dimension - for generating such non-Kochen-Specker hypergraphs in any
dimension and give examples in up to 16-dim spaces. Our automated generation is
probabilistic and random, but the statistics of accumulated data enable one to
filter out sets with the required size and structure.
