On defining sets of full designs

A defining set of a t-(nu. k. lambda) 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 {vertical bar M vertical bar vertical bar M is a minimal defining set of D}. The unique simple design with parameters 2-(nu, k, ((nu-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. (C) 2010 Elsevier B.V. All rights reserved