Root cases of large sets of \(t\)-designs (Q1869219)

A large set of \(t\)-\((v,k,i)\) 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 \(t\)-\((v,k,i)\) designs, where \(N = (v-t)!/[(k-t)!(v-k)!i]\). The authors extend some of the recursive methods for constructing large \(t\)-designs of prime sizes, in order to show that, to construct all possible large sets with the given \(N, t, k\), it suffices to construct a finite number of large sets called by the authors root cases. As a corollary, it is shown that the trivial necessary conditions for the existence of LS[3]\((2,k,v)\) are also sufficient for \(k\) equal to or less than \(80\).