The existence of two pairwise additive \(\mathrm{BIBD}(v, 2, 1)\) for any \(v\). (English) Zbl 1418.05027
Summary: The existence of additive balanced incomplete block (BIB) designs has been discussed with direct and recursive constructions in [M. Sawa et al., J. Comb. Des. 15, No. 3, 235–254 (2007; Zbl 1204.05029)]. In this article, pairwise additive BIB designs are proposed and then new recursive methods of constructing such designs are provided. It is finally shown that for any integer \(v\), two pairwise additive \(\mathrm{B}(v, 2, 1)\) can be constructed. As a by-product we present a recursive construction of multiply nested BIB designs of block sizes \(2^\ell\) for some \(\ell\).
MSC:
| 05B05 | Combinatorial aspects of block designs |
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 62K10 | Statistical block designs |
Keywords:
incidence matrix; balanced incomplete block (BIB) design; additive BIB design; pairwise additive BIB design; nested BIB design; multiply nested BIB design; ordered design; perpendicular arrayCitations:
Zbl 1204.05029References:
| [1] | Bierbraner, J.; Colbourn, C. J (ed.); Dinitz, J. H (ed.), Ordered designs, perpendicular arrays, and permutation sets, 543-547 (2007), Boca Raton, FL |
| [2] | Hanani, H., The existence and construction of balanced incomplete block designs, Ann. Math. Stat., 32, 361-386 (1961) · Zbl 0107.36102 · doi:10.1214/aoms/1177705047 |
| [3] | Matsubara, K.; Sawa, M.; Matsumoto, D.; Kiyama, H.; Kageyama, S., An addition structure on incidence matrices of a BIB design, Ars Combin., 78, 113-122 (2006) · Zbl 1164.05316 |
| [4] | Morgan, J. P.; Preece, D. A.; Rees, D. H., Nested balanced incomplete block designs, Discrete Math., 231, 351-389 (2001) · Zbl 0988.05018 · doi:10.1016/S0012-365X(00)00332-0 |
| [5] | Preece, D. A., Nested balanced incomplete block designs, Biometrika, 54, 479-486 (1976) · doi:10.1093/biomet/54.3-4.479 |
| [6] | Raghavarao, D. 1988. Constructions and combinatorial problems in design of experiments, New York, NY, Dover. · Zbl 0731.62133 |
| [7] | Sawa, M.; Matsubara, K.; Matsumoto, D.; Kiyama, H.; Kageyama, S., The spectrum of additive BIB designs, J. Combin. Des., 15, 235-254 (2007) · Zbl 1204.05029 · doi:10.1002/jcd.20147 |
| [8] | Sawa, M.; Kageyama, S.; Jimbo, M., Compatibility of BIB designs, Stat. Appl., 6, 73-89 (2008) |
| [9] | Sawa, M.; Matsubara, K.; Matsumoto, D.; Kiyama, H.; Kageyama, S., Decomposition of an all-one matrix into incidence matrices of BIB designs, J. Stat. Appl., 4, 455-464 (2009) · Zbl 1295.62075 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
