Constructions and bounds for (m,3)-uniform splitting system

Abstract

Suppose $m$ and $t$ are integers such that $0 < t \leq m$. An $(m,t)$-splitting system is a pair $(X, \mathbf{B})$ where $|X|=m$, $\mathbf{B}$ is a set of subsets of $X$, called blocks such that for every $Y \subseteq X$ and $|Y|=t$, there exists a block $B \in \mathbf{\mathbf{B}}$ such that $|B \cup Y| = \lfloor t/2 \rfloor$. An $(m,t)$-splitting system is uniform if every block has size $\lfloor m/2 \rfloor$. We will give some results on uniform splitting systems for $t=3$.