DOI link for Symmetric Designs
Symmetric Designs book
Omitting the restriction to short orbits, Laue (unpublished) reports that there are at least 719,144 nonisomorphic S(5, 6, 108)s and at least 945 nonisomorphic S(5, 6, 132)s.
See Also §II.3 Steiner 2-designs are BIBDs with index one. §II.4 S(t, k, v) are t-designs with index one. §VI.63 t-wise balanced designs are generalizations of Steiner systems. §VII.2 Finite geometries furnish many examples of S(t, k, v).  Textbook covering most of the material in this chapter.  A comprehensive monograph on triple systems.  Resolvable t-designs.  A comprehensive survey of SQS.  Necessary conditions for Steiner systems.  Asymptotically most efficient known isomorphism algorithm for
S(2, k, v)s.