Partition of ordered triples into uniform holey ordered designs

Abstract

A large set $\text{LOD}\left(v\right)$ is a partition of all ordered triples of a $v$-set into $v-2$ disjoint ordered designs of order $v$. In this paper, we generalize the large set $\text{LOD}\left(v\right)$ with $v=gt$ to the notion of $\text{POT}\left(g^t\right)$, representing a partition of all ordered triples of a $gt$-set into disjoint uniform holely ordered designs $\text{HOD}\left(g^t\right)$s. We show that a $\text{POT}\left(g^t\right)$ exists if and only if $g=1,2$ and $t\ge 3$, except for $\left(g,t\right)=\left(1,6\right)$. Moreover, we study the existence of a $\text{POT}\left(g^t\right)$ with every member $\text{HOD}\left(g^t\right)$ having a kind of resolution. We show that a resolvable $\text{POT}\left(g^t\right)$ exists if and only if $g=1,2$, $t\ge 3$, $\left(g,t\right)\ne \left(1,6\right)$, with 27 possible exceptions. For almost resolvable $\text{POT}\left(2^t\right)$s, we prove the asymptotic existence and present a few infinite families.