Existence of OBIBDs with k=4 with and without nesting Original Research Article

Methods developed for the recursive construction of sets of balanced incomplete block designs with adjusted orthogonality (OBIBDs) are applied to designs with block size 4, with 2 sets of treatments. The methods are extended to include designs with blocks of size 4 nested into sub-blocks of size 2. We also complete the existence results of Greig and Rees for OBIBDs with block size 3 and 2 sets of treatments, showing the necessary conditions are sufficient for existence. Some pairwise balanced design closures are given, with 181 possible exceptions for K={8,9}, 29 possible exceptions when K is the set of prime powers equivalent to 0,1 (mod 8), and when K is the set of prime powers equivalent to 0,1 (mod 4) excluding 4 and 5 we list 26 possible exceptions under the extra restriction v≢4 (mod 8).