The paper considered the construction of BIBDs using potential Lotto Designs (LDs) earlier derived from qualifying parent BIBDs. The study utilized Li’s condition \u0002 pr t−1 \u0003 ( t−1 2 ) + \u0002pr−\u0002 pr t−1 \u0003(t−1) 2 \u0003 < ( p 2 ) λ, to determine the qualification of a parent BIBD (v, b, r, k, λ) as LD (n, k, p, t) constrained on v ≥ k, v ≥ p, t ≤ min{k, p} and then considered the case k = t since t is the smallest number of tickets that can guarantee a win in a lottery. The (15, 140, 28, 3, 4) and (7, 7, 3, 3, 1) BIBDs were selected as parent BIBDs to illustrate the procedure. These BIBDs yielded three potential LDs each. Each of the LDs was completely generated and their properties studied. The three LDs from the (15, 140, 28, 3, 4) produced (9, 84, 28, 3, 7), (10, 120, 36, 3, 8) and (11, 165, 45, 3, 9) BIBDs while those from the (7, 7, 3, 3, 1) produced the (5, 10, 6, 3, 3), (6, 20, 10, 3, 4) and (7, 35, 15, 3, 5) BIBDs. The produced BIBDs follow the generalization (v + 1, b + r + λ + 1, r +λ+1, k, λ+1) where (v, b, r, k, λ) are the parameters of the (9, 84, 28, 3, 7) and (5, 10, 6, 3, 3) BIBDs. All the BIBDs produced are unreduced designs.<\/p>\r\n","references":"[1] K. Hinkelmann and O. Kempthorne, Design and Analysis of Experiments.\r\nVolume 2: Advanced Experimental Design. John Wiley & Sons, Inc, 2005.\r\n[2] C. J. Colbourn and J. H. Dinitz, \"Block designs,\" in Handbook of Discrete and Combinatorial Mathematics, K. H. Rosen, J. G. Michael,\r\nJ. L. Gross, J. W. Grossman, and D. R. Shier., Eds. Boca Raton: CRC Press, 2000, pp. 793-804.\r\n[3] A. P. Street and D. J. Street, Combinatorics of Experimental Design.\r\nOxford: Clarendon Press, 1987.\r\n[4] T. Beth, D. Jungnickel, and H. Lenz, Design Theory, 2nd ed. Cambridge\r\nUniversity Press, 1999.\r\n[5] S. Prestwich, \"A local search algorithm for balanced incomplete block\r\ndesigns,\" in Lecture Notes in Computer Science. Springer 2627, 2003, pp. 132-143.\r\n[6] D. Yokoya and T. Yamada, \"A tabu search algorithm to construct bibds\r\nusing mip solvers,\" in The Eighth International Symposium on Operation\r\nResearch and its Applications (ISORA-09), 20-22nd September, Zhangjiajie, China, 20-22nd September 2009, pp. 179-189.\r\n[7] D. R. Stinson, Combinatorial Designs: Constructions and Analysis.\r\nNew York, Inc.: Springer-Verlag, 2004.\r\n[8] R. Mathon and A. Rosa, \"2-(v, k, \u03bb) designs of small order,\" in Handbook of Combinatorial Designs, C. J. Colbourn and J. H. Dinitz,\r\nEds. Boca Raton: CRC Press, 2007, pp. 25-57.\r\n[9] R. E. Bradley, \"Euler and the genoese lottery,\" 2001, (Online; accessed 15-July-2009). Available: \\url{http:\/\/www.adelphi.\r\nedu\/bradley} \r\n[10] P. C. Li, \"Some results on lotto designs,\" Ph.D. dissertation, Dept. of Computer Science, University of Manitoba, Canada, 1999.\r\n[11] J. A. Bate, \"A generalized covering problem,\" Ph.D. dissertation, Dept.\r\nOf Computer Science, University of Manitoba, Canada, 1978, 118pp.\r\n[12] J. A. Bate and G. H. J. van Rees, \"Lotto designs,\" The Journal of\r\nCombinatorial Mathematics and Combinatorial Computing, vol. 28, pp.\r\n15-39, 1998.\r\n[13] W. R. Grndlingh, \"Two new combinatorial problems involving dominating\r\nsets for lottery schemes,\" Ph.D. dissertation, Department of\r\nApplied Mathematics, University of Stellenbosch, South Africa, 2004,\r\nxxv+ 187pp.\r\n[14] V. Kumar, \"Construction of partially balanced incomplete block designs\r\nthrough unreduced balanced incomplete block designs,\" Journal of Indian Society of Agricultural Statistics, vol. 61, no. 1, pp. 38-41, 2007.\r\n[15] J. K. Baksalary and P. D. Puri, \"Pairwise-balanced, variance-balanced\r\nand resistant incomplete block designs revisited,\" Annals of the Institute\r\nof Statistical Mathematics, vol. 42, no. 1, pp. 163-171, 1990.\r\n[16] K. H. Rosen, Discrete Mathematics and its Applications, 2nd ed. New\r\nYork, USA: McGraw-Hill, 1991.","publisher":"World Academy of Science, Engineering and Technology","index":"Open Science Index 68, 2012"}