On \(3-(8,4,\lambda)\) Designs

The set of all distinct blocks of a \(t\)-design is referred to as the support of the design and its cardinality is denoted by \(b^*\). In this article (i) the set of all possible \(b^*\)’s for the case of \(3\)-\((8,4,\lambda)\) designs is determined and for each feasible \(b^*\) a design with a minimum \(b\) is produced;(ii) it is shown that a \(2\)-\((8,4,3\lambda)\) design is a \(3\)-\((8,4,\lambda)\) design if and only if it is self-complementary; (iii) it is shown that there are at least \(63\) pairwise non-isomorphic \(3\)-\((8,4,5)\) designs.