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

Renwang Su 1 and Jinhua Wang 2
1 College of Statistics and Mathematics, Zhejiang Gongshang University, Hangzhou 310035, P. R. China
2 School of Sciences, Nantong University, Nantong 226007, P. R. China

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 $v\equiv 3$ (mod 6) with $v\geq 51$. As its consequence, some new $(2,w)$-threshold schemes have been obtained.