Existence of OBIBDs with k=4 with and without nesting

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 su7cient 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). c © 2003 Elsevier Science B.V. All rights reserved.