Abstract: When trying to enumerate all BIBD-s for given parameters,
their natural solution space appears to be huge and grows extremely with the number of points of the design. Therefore,
constructive enumerations are often carried out by assuming additional
constraints on design-s structure, automorphisms being mostly used ones. It remains a hard task to construct designs with trivial
automorphism group – those with no additional symmetry – although it is believed that most of the BIBD-s belong to that case. In
this paper, very many new designs with parameters 2-(13, 5, 5), 2-(16, 6, 5) and 2-(21, 6, 4) are constructed, assuming an action of an
automorphism of order 3. Even more, it was possible to construct millions of such designs with no non-trivial automorphisms.