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The Construction of Pairwise Additive Minimal BIB Designs with Asymptotic Results

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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.
Cite this paper
Matsubara, K. and Kageyama, S. (2014) The Construction of Pairwise Additive Minimal BIB Designs with Asymptotic Results. Applied Mathematics, 5, 2130-2136. doi: 10.4236/am.2014.514207. 
 

[1] 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
[2] 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
[3] 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
[4] 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
[5] Matsubara, K. and Kageyama, S. (to be Published) The Existence of 3 Pairwise Additive for Any . Journal of Combinatorial Mathematics and Combinatorial Computing.
[6] 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.
[7] 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
[8] Sawa, M., Kageyama, S. and Jimbo, M. (2008) Compatibility of BIB Designs. Statistics and Applications, 6, 73-89.
[9] 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.
[10] Raghavarao, D. (1988) Constructions and Combinatorial Problems in Design of Experiments. Dover, New York.
[11] 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
[12] Colbourn, C.J. and Rosa, A. (1999) Triple Systems. Oxford Press, New York, 404-406.
[13] Granville, A. (1988) Nested Steiner n-Cycle Systems and Perpendicular Arrays. Journal of Combinatorial Mathematics and Combinatorial Computing, 3, 163-167.                                                 eww141021lx
[14] 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.
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