On Defining sets of Full Designs and of Designs Related to Them

Defining sets of balanced incomplete block designs (BIBDs) were introduced by Ken Gray. Various authors have since identified minimal defining sets of particular BIBDs or classes of BIBDs, usually among those with small values of \( \lambda \).

Here we present results based on defining sets of full designs, that is, designs comprising all \( k \)-tuples on a given set of \( v \) elements. These defining sets are useful, despite their relatively large \( \lambda \) values, since we show that a defining set of any simple BIBD can often be derived from a defining set of the corresponding full design. This leads to an upper bound on the number of simple designs with given parameters, provided that a \( (v,k,\lambda) \) BIBD exists for minimum feasible \( \lambda \).