Beautifully Ordered Balanced Incomplete Block Designs

Abstract

Beautifully Ordered Balanced Incomplete Block Designs, $\text{BOBIBD}(v,k,\lambda ,k_1,{\lambda }_1)$, are defined and the proof is given to show that necessary conditions are sufficient for the existence of BOBIBDs with block size $k=3$ and $k=4$ for $k_1=2$ except possibly for eleven exceptions. Existence of BOBIBDs with block size $k=4$ and $k_1=3$ is demonstrated for all but one infinite family and the non-existence of $\text{BOBIBD}(7,4,2,3,1)$, the first member of the unknown series, is shown.