On defining sets of full designs

A defining set of a t -( v , k , λ ) design is a subcollection of its blocks which is contained in a unique t -design with the given parameters on a given v -set. A minimal defining set is a defining set, none of whose proper subcollections is a defining set. The spectrum of minimal defining sets of a design D is the set { | M | ∣ M is a minimal defining set of D } . The unique simple design with parameters 2 − ( v , k , v − 2 k − 2 ) is said to be the full design on v elements; it comprises all possible k -tuples on a v set. We provide two new minimal defining set constructions for full designs with block size k ≥ 3 . We then provide a generalisation of the second construction which gives defining sets for all k ≥ 3 , with minimality satisfied for k = 3 . This provides a significant improvement of the known spectrum for designs with block size three. We hypothesise that this generalisation produces minimal defining sets for all k ≥ 3 .