Kirkman packing designs KPD ({w,s∗},v) and related threshold schemes

A Kirkman packing design KPD ({w,s∗},v) is a resolvable packing with maximum possible number of parallel classes, each class containing one block of size s and all other blocks of size w. A (t,w)-threshold scheme is a way of distributing partial information (shadows) to w participants, so that any t of them can easily calculate a key, but no subset of fewer than t participants can determine the key. In this paper we improve the existence results on KPD ({3,s∗},v) for s=4,5. We also obtain some results on KPD ({4,s∗},v) for s=5,6. These results can be used to give some new (2,w)-threshold schemes.