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The Construction of Pairwise Additive Minimal BIB Designs with Asymptotic Results
Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan
Hiroshima Institute of Technology, Hiroshima, Japan
- 1 Graduate School of Science, Hiroshima University, Higashi-Hiroshima, Japan
- 2 Hiroshima Institute of Technology, Hiroshima, Japan
Applied Mathematics·Volume 05 (2014)·Pages 2130–2136·Published 28 July 2014·DOI10.4236/am.2014.514207
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Abstract
An asymptotic existence of balanced incomplete block (BIB) designs and pairwise balanced designs (PBD) has been discussed in [1]-[3]. On the other hand, the existence of additive BIB designs and pairwise additive BIB designs with k = 2 and λ = 1 has been discussed with direct and recursive constructions in [4]-[8]. In this paper, an asymptotic existence of pairwise additive BIB designs is proved by use of Wilson’s theorem on PBD, and also for some l and k the exact existence of l pairwise additive BIB designs with block size k and λ = 1 is discussed.
KeywordsIncidence MatrixPairwise Balanced Design (PBD)Balanced Incomplete Block Design (BIBD)Additive BIB DesignPairwise Additive BIB DesignWilson’s Theorem
- Wilson, R.M. (1972) An Existence Theory for Pairwise Balanced Designs I. Journal of Combinatorial Theory, Series A, 13, 220-245. http://dx.doi.org/10.1016/0097-3165(72)90028-3
- Wilson, R.M. (1972) An Existence Theory for Pairwise Balanced Designs II. Journal of Combinatorial Theory, Series A, 13, 246-273. http://dx.doi.org/10.1016/0097-3165(72)90029-5
- Wilson, R.M. (1975) An Existence Theory for Pairwise Balanced Designs III. Journal of Combinatorial Theory, Series A, 18, 71-79. http://dx.doi.org/10.1016/0097-3165(75)90067-9
- Matsubara, K. and Kageyama, S. (2013) The Existence of Two Pairwise Additive for any . Journal of Statistical Theory and Practice, 7, 783-790. http://dx.doi.org/10.1080/15598608.2013.783742
- Matsubara, K. and Kageyama, S. (to be Published) The Existence of 3 Pairwise Additive for Any . Journal of Combinatorial Mathematics and Combinatorial Computing.
- Matsubara, K., Sawa, M., Matsumoto, D., Kiyama, H. and Kageyama, S. (2006) An Addition Structure on Incidence Matrices of a BIB Design. Ars Combinatoria, 78, 113-122.
- Sawa, M., Matsubara, K., Matsumoto, D., Kiyama, H. and Kageyama, S. (2007) The Spectrum of Additive BIB Designs. Journal of Combinatorial Designs, 15, 235-254. http://dx.doi.org/10.1002/jcd.20147
- Sawa, M., Kageyama, S. and Jimbo, M. (2008) Compatibility of BIB Designs. Statistics and Applications, 6, 73-89.
- Mullin, R.C. and Gronau, H.D.O.F. (2007) PBDs and GDDs: The Basics. In: Colbourn, C.J. and Dinitz, J.H., Eds., The CRC Handbook of Combinatorial Designs, 2nd Edition, CRC Press, Boca Raton, 160-193.
- Raghavarao, D. (1988) Constructions and Combinatorial Problems in Design of Experiments. Dover, New York.
- Colbourn, C.J. and Ling, A.C.H. (1997) Pairwise Balanced Designs with Block Sizes 8, 9 and 10. Journal of Combinatorial Theory, Series A, 77, 228-245. http://dx.doi.org/10.1006/jcta.1997.2742
- Colbourn, C.J. and Rosa, A. (1999) Triple Systems. Oxford Press, New York, 404-406.
- Granville, A. (1988) Nested Steiner n-Cycle Systems and Perpendicular Arrays. Journal of Combinatorial Mathematics and Combinatorial Computing, 3, 163-167.
- Abel, R.J.R., Colbourn, C.J. and Dinitz, J.H. (2007) Mutually Orthogonal Latin Square. In: Colbourn, C.J. and Dinitz, J.H., Eds., The CRC Handbook of Combinatorial Designs, 2nd Edition, CRC Press, Boca Raton, 160-193.