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

Abstract

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 $b^{\ast }$. In this article (i) the set of all possible $b^{\ast }$’s for the case of $3$-$(8,4,\lambda )$ designs is determined and for each feasible $b^{\ast }$ 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.