On defining sets of full designs

A defining set of a tt-(v,k,λv,k,λ) design is a subcollection of its blocks which is contained in a unique tt-design with the given parameters on a given vv-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 DD is the set {|M|∣M{|M|∣M is a minimal defining set of D}D}. The unique simple design with parameters 2−(v,k,v−2k−2) is said to be the full design   on vv elements; it comprises all possible kk-tuples on a vv set. We provide two new minimal defining set constructions for full designs with block size k≥3k≥3. We then provide a generalisation of the second construction which gives defining sets for all k≥3k≥3, with minimality satisfied for k=3k=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≥3k≥3.