Existence of OBIBDs with k = 4 with and without nesting

  • Authors:

  • R. Julian R. Abel;Malcolm Greig;D. H. Rees

  • Affiliations:

  • School of Mathematics, University of New South Wales, Sydney 2052, Australia;Greig Consulting, 317-130 East 11th Street, North Vancouver, BC, Canada V7L 4R3;Institute of Mathematics and Statistics, Cornwallis Building, University of Kent, Canterbury, Kent, CT2 7NF, UK

  • Venue:
  • Discrete Mathematics - Special issue: The 18th British combinatorial conference
  • Year:
  • 2003

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Abstract

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).