Haitao Cao

A perfect $(t,w,v;m)$-threshold scheme is a simple $w$-uniform hypergraph $(\X, \A)$, where $\X$ is a set of $v$ elements (shadows), together with a partition of the block set $\A$ into $m$ parts, say $\A=\{\A_1,\A_2,\ldots,\A_m\}$, such that the following properties are satisfied: (i) if $B\in\A_i$ and $B'\in\A_j$, where $i\not=j$, then $|B\cap B'|< t$.(ii) for any subset $S$ of $t'< t$ shadows, there exists a non-negative integer $\lambda(S)$ such that for every $i\ (1\le i\le m)$ there are exactly $\lambda(S)$ blocks $B$ such that $S\subseteq B\in \A_i$. Let M$(t,w,v)$ denote the maximum value of $m$ such that there exists a perfect $(t,w,v;m)$-threshold scheme. It is well known that $M(t,w,v)\le (v-t+1)/(w-t+1)$. Threshold schemes were first introduced by Shamir and Blakely in 1979. Since then, many constructions have been given for threshold schemes. In 1988, Stinson and Vanstone gave some constructions for perfect threshold schemes meeting the upper bound using certain combinatorial designs, called partitionable Steiner systems S$(t,w,v)$. A partitionable Steiner system S$(3,3,v)$ is also called a large set of Steiner triple system LSTS(v) with $v\equiv\ 1,3\ (\mod 6)$. In 1989, Schellenberg and Stinson used large sets of packings LSP$(2,3,v)$ to construct these perfect threshold schemes for $v\equiv\ 0,2,4,5\ (\mod 6)$. The existence problem for LSP$(2,3,v)$ has been solved with a few exceptions by many people. In this paper, we shall use large sets of packings LSP$(2,4,v)$ to construct perfect threshold schemes and give more accurate upper bounds for M$(3,4,v)$.