On Kirkman Packing Designs \(KPD({3, 4^*, 5^*}, v)\)s

Abstract

A Kirkman packing design \(KPD({w, s^*, t^*}, v)\) is a Kirkman packing with maximum possible number of parallel classes, such that each parallel class contains one block of size \(s\), one block of size \(t\) and all other blocks of size \(w\). A \((k, w)\)-threshold scheme is a way of distributing partial information (shadows) to \(w\) participants, so that any \(k\) of them can determine a key easily, but no subset of fewer than \(k\) participants can calculate the key. In this paper, the existence of a \(KPD({3, 4^*, 5^*}, v)\) is established for every \(v \equiv 3 \pmod{6}\) with \(v \geq 51\). As its consequence, some new \((2, w)\)-threshold schemes have been obtained.