The Spectrum for Self-Converse Directed $BIBDs$ with Block Size Four

Abstract

A directed balanced incomplete block design ($DB$($k$, $\lambda$;$v$)) $(X,\mathcal{B})$ is called self-converse if there is an isomorphic mapping $f$ from $(X,\mathcal{B})$ to $(X,{\mathcal{B}}^{-1})$, where ${\mathcal{B}}^{-1}=\{B^{-1}:B\in \mathcal{B}\}$ and $B^{-1}=(x_k,x_{k-1},\dots ,x_2,x_1)$ for $B=(x_1,x_2,\dots ,x_{k-1},x_k)$. In this paper, we give the existence spectrum for self-converse $DB$($4$,$\lambda$;$v$) for any $\lambda \ge 1$.