All Admissible $3-(v,4,\lambda )$ Directed Designs Exist

Abstract

In a $t-(v,k,\lambda )$ directed design, the blocks are ordered $k$-tuples and every ordered $t$-tuple of distinct points occurs in exactly $\lambda$ blocks (as a subsequence). We show that a simple $3-(v,4,2)$ directed design exists for all $v$. This completes the proof that the necessary condition $\lambda v\equiv 0(\mathrm{mod}2)$ for the existence of a $3-(v,4,\lambda )$ directed design is sufficient.