\(t\)-designs with few intersection numbers (Q1175994)

The authors present a method to obtain new \(i\)-designs from \(t\)-designs with few intersection numbers provided some numerical conditions are met. Specifically, let \(B\) be a fixed block of a \(t-(v,k,\lambda)\) design having \(j\) distinct intersection numbers. Repeated blocks are allowed. Let \(x\) be an intersection number. If \(i+j-1\leq t\) and \(i\leq v-k\) then the incidence structure whose point set is \(B\) complement and whose blocks are those of the original design which intersect \(B\) in exactly \(x\) points is an \(i\)-design (possibly with repeated blocks). Moreover, the new design has no repeated blocks if the original did not and \(x=0\), or if \(\lambda=1\) and \(k-x\geq t\). The authors hope that this construction may help in classifying quasi-symmetric 3-designs with positive intersection numbers.