Further Existence Results on Beautifully Ordered Balanced Incomplete Block Designs

Abstract

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($v,4,2,3,1$), $v\equiv 1(\mathrm{mod}6)$, is demonstrated for all $v\ge 19$.