Beautifully Ordered Balanced Incomplete Block Designs

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.