Handbook of Combinatorial Designs, Second Edition

New Results in Part IV

This page contains new results in the area of combinatorial designs that have occurred since the publication of the Handbook of Combinatorial Designs, Second Edition in November 2006. The results here would be contained in Part IV of the Handbook.

Last edited 8/16/10

Page 256, Theorem 4.5. This theorem is now a special case of the following theorem: Let u, v and x be positive integers with vu. Then a 3-GDD of type u1 v1 1x exists if and only if u, v and x are odd, uv+ux+vx+\binom{x}{2} ≡ 0 (mod 3), and xu. This result appears in Darryn Bryant and Daniel Horsley, Steiner triple systems with two disjoint subsystems, J. Combin. Des., 14, 14-24 (2006). Darryn Bryant (db@maths.uq.edu.au) Sept. 2007.

Page 260, Theorem 4.32. The four possible exceptions with m = 3 and λ = 1, namely (n,t) \in {(6,14), (6,15), (6,18), (6,23)}, were constructed directly by Cao, Wang and Wei. The result appears in H. Cao, L. Wang, R. Wei, The existence of HGDDs with block size four and its application to double frames, Discrete Mathematics, Volume 309, Issue 4 (2009), Pages 945-949. Fei Gao (feigao.chn@gmail.com), March 2010.

Page 265, Theorem 5.44. The 4-RGDD of type 1227 was found in Theorem 2.1 of X. Sun and G. Ge, "Resolvable group divisible designs with block size four and general index," Discrete Math. vol. 309, pp. 2982-2989, 2009. http://dx.doi.org/10.1016/j.disc.2008.07.029. Fei Gao (feigao.chn@gmail.com), August 2010.

Page 265, Theorem 5.44. 4-RGDD of types 2142 , 2346, 654 and 2423 were found in E. Schuster and G. Ge, "On uniformly resolvable designs with block sizes 3 and 4", Des. Codes Cryptogr., vol. 57, pp 45 -- 69, 2010. http://dx.doi.org/10.1007/s10623-009-9348-1. Fei Gao (feigao.chn@gmail.com), August 2010.

Page 265, Theorem 5.45. The {4,3}-RGDD of type 254 was found in Theorem 2.2 of X. Sun and G. Ge, "Resolvable group divisible designs with block size four and general index," Discrete Math. vol. 309, pp. 2982-2989, 2009. http://dx.doi.org/10.1016/j.disc.2008.07.029. Fei Gao (feigao.chn@gmail.com), August 2010.

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