A combinatorial design can be described as a very regular finite structure consisting of two kinds of objects, points and blocks. The solution space of a combinatorial structure grows enormously with the number of points, which is known as the combinatorial explosion. Since a complete classification of such a structure is generally not possible, a construction is attempted by assuming additional constraints, which the structure could contain and which are often formulated as some automorphism group action. Designs having the trivial order of automorphism groups are of special interest since most of designs are of such kind, but only few are known, because of the particularly complex construction. This work deals with development of efficient algorithms for t-designs construction, based on tactical decomposition which could be achieved from an action of a prim order automorphism, with the possibility to use or to forget the group action in this construction step. Among the symmetric designs, partial classification for parameters (36, 15, 6) and (41, 16, 6), assuming an action of an automorphism of order 3, was realized. The same partial classification was done for non-symmetric design with parameters (13, 5, 5). Very many new non-symmetric designs were constructed for parameter triples (16, 6, 5) and (21, 6, 4). The generality of the algorithm was shown for small parameters t-(v, k, ?), t > 2. The number of new designs constructed here can be counted in hundreds of millions, and also over a million of designs having no non-trivial automorphisms was achieved.