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 \(B^{-1} = (x_k,x_{k-1},\ldots,x_2,x_1)\) for \(B=(x_1,x_2,\ldots,x_{k-1},x_{k})\). In this paper, we give the existence spectrum for self-converse \(DB\)(\(4\),\(\lambda\);\(v\)) for any \(\lambda \geq 1\).