Further Existence Results on Beautifully Ordered Balanced Incomplete Block Designs

Beautifully Ordered Balanced Incomplete Block Designs, BOBIBD( \( v, k, \lambda, k_1, \lambda_1 \) ), were introduced by Chan and Sarvate along with some existence results for block size \( 3 \) and \( 4 \). We have shown that necessary conditions are sufficient for the existence of BOBIBDs with \( k = 5 \) for \( k_1 = 2 \) and \( 3 \) along with partial results for \( k_1 = 4 \). We also claim the nonexistence of cyclic solutions for certain BOBIBDs. The existence of the previously unknown BOBIBD(\( v, 4, 2, 3, 1 \)), \( v \equiv 1 \pmod{6} \), is demonstrated for all \( v \geq 19 \).