Directed \(B(K,1;v)\) with \(K= \{4,5\}\) and \(\{4,6\}\) related to deletion/insertion-correcting codes (Q2713362)

A directed pairwise balanced design DB(\(K\),1;\(v\)), where \(v\) is an integer and \(K\) is a set of integers, is a collection of totally ordered \(k\)-subsets, called blocks, \(k \in K\), of a \(v\)-set \(X\), such that every ordered pair of distinct elements of \(X\) occurs in exactly one block (in the correct order). Directed pairwise balanced designs have applications in the construction of \(t\)-deletion/insertion-correcting codes. The authors prove that, for \(v \geq 4\), a DB(\{4,5\},1;\(v\)) exists if and only if \(v \notin \{6,8,9,12,14\}\), and a DB(\{4,6\},1;\(v\)) exists if and only if \(v \equiv 0\) or \(1 \bmod 3\) and \(v \notin \{9,15\}\). The spectra for DB(\{\(k\)\},1;\(v\)), \(3 \leq k \leq 6\), had been previously determined by various authors.