Some Constructions of Block Designs

Abstract

Using a blend of Drake’s and Saha’s techniques, we construct a $\text{BTD}(n^2/4;(n^2+n)/2;2n–4,3,2n+2;n;8)$ whenever $n$ is a power of $2$, as well as some new symmetric $\text{BTDs}$.It is known that the necessary condition $v\equiv 1(\mathrm{mod}2)$ is sufficient for the existence of simple $\text{BIBD}(v,3,3)$.In the second part of this paper, we provide a simple construction based on graph factorization to prove this result whenever $v$ is not divisible by $3$.We then expand upon this result to exhibit further constructions of $\text{BTDs}$.