On the Support Size of \(3\)-Designs with Repeated Blocks

The set of all distinct blocks of an \(t\)-(v,k) design is referred to as the support of the design, and its cardinality is denoted by \(b^*\). By generalizing a method on BIB designs called “trade off” to \(3\)-designs, a table for \(3\)-(9,4) designs with each \(60 \leq b^* \leq 126 = {\binom{9}{4}}\) is constructed. Also, we have produced over 2500 non-isomorphic \(3\)-(9,4) designs with \(\lambda = 6\). By utilizing this generalized trade off method along with two other methods, a table for \(3\)-(10,4) designs with 156 different \(b^*\)’s is constructed. By a recursive lower bound on the minimum value of \(b^*\) in all \(t\)-(v,k) designs, it is shown that \(b^*_{min}[3-(9,4)] \geq 36,\) and \(b^*_{min}[3\)-(10,4)] = 30.