  The rate regions of multilevel diversity coding systems (MDCS), a sub-class
of the broader family of multi-source multi-sink networks with special
structure, are investigated. After showing how to enumerate all non-isomorphic
MDCS instances of a given size, the Shannon outer bound and several achievable
inner bounds based on linear codes are given for the rate region of each
non-isomorphic instance. For thousands of MDCS instances, the bounds match, and
hence exact rate regions are proven. Results gained from these computations are
summarized in key statistics involving aspects such as the sufficiency of
scalar binary codes, the necessary size of vector binary codes, etc. Also, it
is shown how to generate computer aided human readable converse proofs, as well
as how to construct the codes for an achievability proof. Based on this large
repository of rate regions, a series of results about general MDCS cases that
they inspired are introduced and proved. In particular, a series of embedding
operations that preserve the property of sufficiency of scalar or vector codes
are presented. The utility of these operations is demonstrated by boiling the
thousands of MDCS instances for which binary scalar codes are insufficient down
to 12 forbidden smallest embedded MDCS instances.
