  Quantum contextuality turns out to be a necessary resource for universal
quantum computation and important in the field of quantum information
processing. It is therefore of interest both for theoretical considerations and
for experimental implementation to find new types and instances of contextual
sets and develop methods of their optimal generation. We present an arbitrary
exhaustive hypergraph-based generation of the most explored contextual
sets---Kochen-Specker (KS) ones---in 4, 6, 8, 16, and 32 dimensions. We
consider and analyse twelve KS classes and obtain numerous properties of
theirs, which we then compare with the results previously obtained in the
literature. We generate several thousand times more types and instances of KS
sets than previously known. All KS sets in three of the classes and in the
upper part of a fourth are novel. We make use of the MMP hypergraph language,
algorithms, and programs to generate KS sets strictly following their
definition from the Kochen-Specker theorem. This approach proves to be
particularly advantageous over the parity-proof-based ones (which prevail in
the literature), since it turns out that only a very few KS sets have a parity
proof (in six KS classes 0.01% and in one of them 0%). MMP hypergraph formalism
enables a translation of an exponentially complex task of solving systems of
nonlinearequations, describing KS vector orthogonalities, into a statistically
linearly complex task of evaluating vertex states of hypergraph edges, thus
exponentially speeding up the generation of KS sets and enabling us to generate
billions of novel instances of them. The MMP hypergraph notation also enables
us to graphically represent KS sets and to visually discern their features.
