Some Constructions of Block Designs

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 \(v \equiv 1 \pmod{2}\) 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 \(\text{BTDs}\).