Extending large sets of \(t\)-designs (Q1924242)

A large set of disjoint \(t\)-designs, denoted here \(\text{LS}(1/n;t,k,v)\), is a partition of the family of \(k\)-subsets of a \(v\)-set into \(n\) equal families each of which is a \(t\)-design. The author presents some recursive constructions for large sets of \(t\)-designs. Among other results, he shows that if an \(\text{LS}(1/p;t,n,u-1)\) exists, for \(p\) prime, and \(np<k<(n+1)p\), then an \(\text{LS}(1/p;t+1,k,pu)\) also exists; starting with a known \(\text{LS}(1/2;6,7,8m-2)\) he thus obtains an \(\text{LS}(1/2;t,2^{t-3}-1,m2^{t-3}-2)\) for \(t\geq 6\) and \(m\geq 2\).