Constructing \(t\)-designs from \(t\)-wise balanced designs (Q866514)

Let \(X\) be a finite set of nonzero cardinality \(v\) and \(B\) be a nonempty collection of subsets of \(X\). If each subset of \(X\) with cardinality \(t>0\) is contained in exactly \(\lambda>0\) of the elements of \(B\), then this structure is called a \(t\)-wise balanced design. This paper provides a procedure for constructing from such a design, a balanced design (called a \(t\)-design) with all the elements of \(B\) having the same cardinality. It is shown that the automorphism group of the new design contains as a subgroup, the automorphism group of the original design. The construction is also used to produce a \(t\)-design for a smaller set (\(X\) with one point removed). The authors state that their construction was found as a result of looking for 2-designs with repeated blocks to help fill up the catalogue by \textit{D. A. Preece} [A selection of BIBDs with repeated blocks, \(r \leq 20\), \(\text{gcd}(b,r,\lambda)=1\), 2003 (Preprint)]. They also provide examples constructed and tested with the DESIGN package in GAP (http://www.gap-system.org, 2004).