Handbook of Combinatorial Designs

New Results

This page contains new results in the area of combinatorial designs that have occurred since the publication of the first edition of the CRC Handbook of Combinatorial Designs . These results are generally now included in the Handbook of Combinatorial Designs, Second Edition which was published in November 2006. This page is maintained by Jeff Dinitz.

The results given on this page are results that improve upon a currently existing table entry or theorem in The CRC Handbook of Combinatorial Designs. We give the result and the page number of the table or theorem where it would occur in the Handbook. We also give the name (and e-mail address) of the mathematician who has reported the new result and the date it was put on this page if no date occurs, it is before March 1, 1997)..

Please help us keep this page up-to-date. Send your new results to either of the editors for inclusion on this page. Remember that the criterion for inclusion is that the result must improve upon a result currently given in the Handbook.

We do not accept responsibility for the correctness of these results. Please contact the person who has reported the result to obtain more information.

This page has grown so much that I have separated it onto several different web pages. Follow the links to the Part that you are interested in.

Part I: Balanced Incomplete Block Designs and t-Designs (pages 1-94)

Part II: Latin Squares, MOLS, and Orthogonal Arrays (pages 95-182)

Part III: Pairwise Balanced Designs (pages 183-226)

Part IV: Other Combinatorial Designs (pages 227-514)

Part V: Applications

Page 578, Reference [18]. This paper has appeared. The reference is: Rodney, P. The existence of interval-balanced tournament designs. J. Combin. Math. Combin. Comput. 19 (1995), 161--170.

Page 582, Example 8.19. Uenal Mutlu (bm373592@muenchen.org) has a construction that shows that L(49,6,6,3) <= 168. A description of the result can be found at http://www.tuco.de/my168a.txt. He has some other interesting lottery information at his site http://www.tuco.de/math1.htm.

Page 582, Example 8.19. Salvatore Minacapelli has a construction that shows that L(49,6,6,3)<=165. This is obtained by combining a L(22,6,3,3)=77 design with his new result L(27,6,4,3)<=88. This result can also be found at http://lottery.merseyworld.com/Wheel/Wheel.html Salvatore Minacapelli (sminacapelli@hotmail.com). April 2000.

Part VI: Related Mathematics and Computational Methods

Page 648, Table 3.13. The following entries can be added to this table:

k = 4, n = 17: 86221634,
k = 4, n = 18: 985870522,
k = 5, n = 16: 2585136675,
k = 6, n = 14: 21609300,
k = 6, n = 15: 1470293675,
k = 7, n = 14: 21609301.
Reported by Anton Betten (anton@btm2xf.mat.uni-bayreuth.de).

Page 664, Theorem 4.39. Extending a result by A. Hartman and A. Rosa (European J. Combin. 6 (1985), no. 1, 45--48), Buratti proved that for any abelian group G of even order, except for G = Z_{2^n} with n>2 , there exists a one-factorization of the complete graph admitting G as a sharply-vertex-transitive automorphism group. M. Buratti, Abelian 1-factorization of the complete graph. European J. Combin. 22 (2001), no. 3, 291--295.

Page 664, Theorem 4.42. Perfect 1-factorizations exist for the values 24390, 50654, 78126, 79508, 103824, 148878, 161052, 300764, 357912, 493040, 571788, 1092728, 1225044. Volker Leck (volker.leck@mail1.uni-rostock.de) March 2001. Also for the values 433, 529, 593, 625, 2809, 3481, 3721, 4489, 6889. Volker Leck, April 2001.

Page 664, Theorem 4.42. Perfect 1-factorizations exist for the values (add one to the following) 529, 2809, 4489, 6889, 11449, 11881, 15625, 22201, 24389, 24649, 26569, 50653, 78125, 79507, 103823, 161051, 205379, 300763, 357911, 371293, 493039, 571787. I. M. Wanless, Atomic Latin squares based on cyclotomic orthomorphisms, (submitted). October 2004.

Page 664, Theorem 4.45. NP(36) >= 12 and NP(40) >= 4. Reference: D. Pike and N. Shalaby, The use of skolem sequences to generate perfect one-factorizations, Ars Combinatoria 59 (2001), pp. 153-159. July 2001.

Page 672, parameters (57,24,11,9). The first column can be given the value 11,084,874,829. Reference: P. Kaski and P.R.J. Östergård, The Steiner triple systems of order 19, Math. Comp. 73 (2004), no. 248, 2075--2092. (patric.ostergard@hut.fi), December 2001.

Page 724, Remarks 9.13. A succesful backtracking search using an orderly algorithm found all Steiner triple systems or order 19: P. Kaski and P.R.J. Östergård [The Steiner triple systems of order 19, Math. Comp. 73 (2004), no. 248, 2075--2092.] Patric Östergård (patric.ostergard@hut.fi), December 2001.

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