On The Spectrum of Minimal Defining Sets of Full Designs

Abstract

A defining set of a t-(v, k, λ) design is a subcollection of the block set of the design which is not contained in any other design with the same parameters. A defining set is said to be minimal if none of its proper subcollections is a defining set. A defining set is said to be smallest if no other defining set has a smaller cardinality. A t-(v, k, λ) design ${D = (V, \fancyscript{B})}$ is called a full design if ${\fancyscript{B}}$ is the collection of all possible k-subsets of V. Every simple t-design is contained in a full design and the intersection of a defining set of a full design with a simple t-design contained in it, gives a defining set of the corresponding t-design. With this motivation, in this paper, we study the full designs when t = 2 and k = 3 and we give several families of non-isomorphic minimal defining sets of full designs. Also, it is proven that there exist values in the spectrum of the full design on v elements such that the number of non-isomorphic minimal defining sets on each of these sizes goes to infinity as v→ ∞. Moreover, the lower bound on the size of the defining sets of the full designs is improved by finding the size of the smallest defining sets of the full designs on eight and nine points. Also, all smallest defining sets of the full designs on eight and nine points are classified.