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The Construction of Pairwise Additive Minimal BIB Designs with Asymptotic Results

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DOI: 10.4236/am.2014.514207    2,043 Downloads   2,374 Views   Citations

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 and k the exact existence of l pairwise additive BIB designs with block size k and λ = 1 is discussed.

Conflicts of Interest

The authors declare no conflicts of interest.

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.

References

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[2] Wilson, R.M. (1972) An Existence Theory for Pairwise Balanced Designs II. Journal of Combinatorial Theory, Series A, 13, 246-273.
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[3] Wilson, R.M. (1975) An Existence Theory for Pairwise Balanced Designs III. Journal of Combinatorial Theory, Series A, 18, 71-79.
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[5] Matsubara, K. and Kageyama, S. (to be Published) The Existence of 3 Pairwise Additive for Any . Journal of Combinatorial Mathematics and Combinatorial Computing.
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