Communicated by L. Teirlinck.
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The paper concerns a study of our previous general construction for simple t-designs, called the basic construction, with the goal to establish existence theorems for t-designs. As a general framework the basic construction involves a great deal of possibilities of combining ingredient designs, and thus computations are necessary for constructing designs by this method. The work shows the results of an investigation finding specified conditions under which the required computations can be avoidable. They thus lead to existence theorems for simple t-designs and many of them have been found. Also a large number of examples are included to illustrate the results.