Block disjoint difference families for Steiner triple systems: \(v\equiv 3 \mod 6\)
Given a group \(G\) of order \(v\), a family of pairwise disjoint \(k\)-tuples of elements from \(G\) is a block disjoint \((v,k,\lambda)\) difference family if the collection of orbits of the \(k\)-tuples under the action of \(G\) form a BIBD\((v,k,\lambda)\). In 1997 the first author and \textit{P. Rodney} [Util. Math. 52, 153-160 (1997; Zbl 0892.05012)] showed that there exists a block disjoint \((v,3,1)\) difference family for all \(v \equiv 1 {\pmod 6}\). In this companion paper the authors use similar techniques to establish the existence of a cyclic block disjoint \((v,3,1)\) difference family for all \(v \equiv 3 {\pmod 6}\) with \(v \geq 3\) and \(v \neq 9\). In the case \(v = 9\), there is a \(1\)-rotational block disjoint \((9,3,1)\) difference family (with point set \(Z_8 \cup \infty\)), namely \(\{\{\infty,0,4\},\{1,2,7\}\}\). Combining this with the result from the companion paper, it is established that a block disjoint \((v,3,1)\) difference family exists for all \(v \equiv 1,3 {\pmod 6}\) with \(v \geq 3\).
scientific article; zbMATH DE number 1802341