Root cases of large sets of t-designs Original Research Article

A large set of t-(v,k,λ) designs of size N, denoted by LS[N](t,k,v), is a partition of all k-subsets of a v-set into N disjoint t-(v,k,λ) designs, where N=(v−tk−t)/λ. A set of trivial necessary conditions for the existence of an LS[N](t,k,v) is N|v−ik−i for i=0,…,t. In this paper we extend some of the recursive methods for constructing large sets of t-designs of prime sizes. By utilizing these methods we show that for the construction of all possible large sets with the given N,t, and k, it suffices to construct a finite number of large sets which we call root cases. As a result, we show that the trivial necessary conditions for the existence of LS[3](2,k,v) are sufficient for k⩽80.