On arcs in quadruple systems (Q2770180)

Let \(B(2,k,v)\) denote a family \(R\) of \(k\)-element subsets (called blocks) of a \(v\)-element set \(V\) such that every two-element subset of \(V\) belongs exactly to one block of \(R\). Such a family is called a Steiner system. An \(s\)-arc in \(B(1,k,v)\) is an \(s\)-element subset of \(V\) such that no block of \(R\) is a subset of the \(s\)-arc. So far arcs in \(B(2,3,v)\) Steiner triples have been considered in literature. In this paper arcs in quadruple systems were dealt with. Also the arcs have been constructed. Further, while calculating the number of arcs in the system of blocks, a table has been given for practical purposes and some unanswered questions were posed for further study.