# New Results in Part II

This page contains new results in the area of combinatorial designs that have occurred since the publication of the CRC Handbook of Combinatorial Designs . The results here would be contained in Part II of the Handbook (pages 95-182).

Last updated 3/14/05

Page 98, Table 1.6. The two question marks can be replaced by the following 23746 (for n=7) and 106228849 (for n=8). It also seems that the numbers in this row of the table are for isomorphism classes of reduced latin squares. Ian Wanless (Ian.Wanless@cs.anu.edu.au). May 2004.

Page 104, Table II.1.12. Values have been computed for n= 11. They are

k L(k,11)
1 1
2 1468457
3 798030483328
4 143968880078466048
5 7533492323047902093312
6 96299552373292505158778880
7 240123216475173515502173552640
8 86108204357787266780858343751680
9 2905990310033882693113989027594240
10 5363937773277371298119673540771840
11 5363937773277371298119673540771840
These numbers were obtained independently by I Wanless and B.D.McKay, but are as yet unpublished. Note that this also affects the table (1.13). Ian Wanless (Ian.Wanless@cs.anu.edu.au). May 2004.

Page 105, example following Remark 1.20. This latin square has precisely 12,265,168 orthogonal mates (see B.M.Maenhaut and I.M.Wanless, Atomic Latin squares of order eleven, JCD 12 (2004), 12--34). Ian Wanless (Ian.Wanless@cs.anu.edu.au). May 2004.

Page 106, Theorem II.1.32. For every odd positive integer m there exists a Latin square or order 3m with no proper Latin subsquares. Combining this with previous results it follows that Latin squares with no proper subsquares exist for all odd orders. Ref: B. M. Maenhaut, I. M. Wanless and B. S. Webb, Subsquare-free Latin Squares of odd order, preprint. October 2004.

Page 117, Theorem II.2.43. Seven MOLS(24) have been found (so the lower bound for N(24) is now 7) by M. Wojtas (wojtas@graf.im.pwr.wroc.pl). June 2003. Click here for the construction.

Page 119, Theorem II.2.50. Six MOLS of side 24 have been found (so the lower bound for N(24) is now 6) by Mieczyslaw Wojtas (wojtas@graf.im.pwr.wroc.pl). April 1999. Click here for the construction.

Page 119, Theorem II.2.50. Eight MOLS of side 36 have been found (so the lower bound for N(36) is now 8) by Mieczyslaw Wojtas (wojtas@graf.im.pwr.wroc.pl). Jan. 2001. Click here for the construction.

Page 119, Theorem II.2.51. Five MOLS of side 39 have been found (so the lower bound for N(39) is now 5) by R. Julian R. Abel (julian@maths.unsw.edu.au). May 1998. Click here for the construction.

Page 121, Theorem II.2.58. Seven MOLS of side 48 have been found (so the lower bound for N(48) is now 7) by Mieczyslaw Wojtas (wojtas@graf.im.pwr.wroc.pl). Sept. 1998. Click here for the construction.

Page 121, Theorem II.2.52. Six MOLS of side 55 have been found (so the lower bound for N(55) is now 6) by Mieczyslaw Wojtas (wojtas@graf.im.pwr.wroc.pl). April 1999. Click here for the construction.

Page 126, Table II.2.72. Four idempotent MOLS of side 15 have been found by D.H. Rees (reported by Julian Abel, julian@maths.unsw.edu.au). January 2002. Click here for the construction.

Page 126, Table II.2.72. Five MOLS of side 54 have been found (so the lower bound for N(54) is now 5) by R. Julian R. Abel (julian@maths.unsw.edu.au). Click here for the construction.

Page 126, Table II.2.72. Five idempotent MOLS of side 55 have been found by R. Julian R. Abel (julian@maths.unsw.edu.au). Click here for the construction.

Page 126, Table II.2.72. Five idempotent MOLS of side 62 have been reported by R. Julian R. Abel (julian@maths.unsw.edu.au). Click here for the construction. March, 2005.

Page 126, Table II.2.72. Seven MOLS of side 75 have been found by R. Julian R. Abel (julian@maths.unsw.edu.au). Click here for the construction. August, 2001.

Page 136, Table II.2.73 As a consequence of the 7 MOLS of order 48 found by M. Wojtas, Julian Abel (julian@maths.unsw.edu.au) and Charlie Colbourn (Charles.Colbourn@uvm.edu) report the following new values for which 7 MOLS exist: 518, 526, 532, 534, 614, 622, 762, 764, and 774. October, 1998.

Page 136, Table II.2.73: As a consequence of the 7 MOLS of order 24 found by M. Wojtas (see above), 7 MOLS of the following orders have been found by Julian Abel (julian@maths.unsw.edu.au), Charlie Colbourn (Charles.Colbourn@uvm.edu), and Mieczyslaw Wojtas (wojtas@mazur.im.pwr.wroc.pl). 7 MOLS now exist for all v > 570 as well for the following orders: 266, 268, 270, 300, 302, 303, 308, 310, 314, 318, 334, 370, 372, 374, 378, 382, 386, 388, 390, 394, 398, 406, 410, 418, 422, 438, 442, 444, 446, 450, 452, 454, 458, 462, 466, 470, 530, 534, 542, 546, 548, 550, 554, 556, 562, 578, 580, 582, 588, 610, 618, 626, 630, 634, 636, 638, 642, 644, 646, 654, 678, 690, 702, 734, 750. June 2003

Page 136, Table II.2.73: As a consequence of 8 MOLS(36) found by Wojtas, Julian Abel (julian@maths.unsw.edu.au) and Charlie Colbourn (Charles.Colbourn@uvm.edu), have established that

• 8 MOLS exist for orders 324, 396, 405, 468, 532, 594, 612, 628, 676, 684, 692, 700, 708, 718, 772, 780, 804, 820, 828, 842, 844, 868, 904, 966, 996, 1012, 1092, 1108, 1116, 1132, 1141, 1188, 1332, 1460, 1462, 1492, 1576, 1580, 1720, 1935, 2068, 2428.
• 8 MOLS imply 7 MOLS for orders 324, 396, 468, 532, 594, 708, 780.
• 7 MOLS exist for orders 332, 356, 364, 380, 420, 428, 460, 476, 500, 508, 516, 564, 572, 660.
June 2003

Page 139, Theorem 2.74. The existence question for r-orthogonal latin squares has been completely settled. The only possible exception that has turned out to be a true exception is the case when n = 6 and r = 36. All other values listed as possible exceptions have been found. (Zhu and Zhang, Completeing the spectrum of r-orthogonal latin squares, Discrete Math, 268 (2003), 343-349). May 2004.

Page 142, Reference 27. Has appeared in the Journal of Combinatorial Theory (A) 70 (1995), pp. 159-164.

Page 144-147, Table II.3.10. Update this table with the following values for N(n;k). Note that these new values will affect some of the entries in the next table (Table II.3.12). Abel and H. Zhang) R. Julian R. Abel (julian@maths.unsw.edu.au). (May 1997 and Sept. 1997) References for these results are: R.J. R. Abel, and F.E.Bennett, Perfect Mendelsohn Designs with block size 7, Discrete Math 190, (1998), 1-14 and R.J. R. Abel, F.E.Bennett and H. Zhang, Perfect Mendelsohn Designs with block size 6 (to appear in JSPI).

 n k N(n;k) 52 6 3 29 2 4 24,30,70 3 4 20,25,28,31,34,40,43,46,77 4 4 30,31,35,36,40,41,45,46,50,51,56 5,6 4 32 5 4 33 6 4 37,43,46 7 4 55 10 4 56 11 4 101 20 4 151 30 4 41, 47, 55 2 5 19, 40,46 3 5 44, 58 5 5 41,51,57 6 5 55 8 5 54 9 5 61 10 5

Page 169, Theorem II.3.14. There are three idempotent incomplete MOLS of order 52 with a hole of size 6, found by R. Julian R. Abel (julian@maths.unsw.edu.au). This leaves (10,1) as the only possible exception for 3 IMOLS.

Page 169, Theorem II.3.14. There are three idempotent incomplete MOLS of order 50 with a hole of size 3, found by R. Julian R. Abel (julian@maths.unsw.edu.au). Click here for the construction. This leaves (10,1) as the only possible exception for 3 idempotent IMOLS. March 2001.

Page 170, Table II.3.21. Several of the HMOLS in this table first appeared in the paper: R.J.R. Abel and H.Zhang, Direct constructions for certain types of HMOLS, Discrete Math. 181 (1998), 1-17. Also, sets of 4 HMOLS exist for the following types: 3^u, u=10, 25, 26; 2^u, u=13, 18, 19; and 619. These HMOLS will appear in R.J. R. Abel, F.E.Bennett and H. Zhang, Perfect Mendelsohn Designs with block size 6 (to appear in JSPI). R. Julian R. Abel (julian@maths.unsw.edu.au). Nov. 1998 and Jan. 1999.

Page 170, Table II.3.21. An entirely updated HMOLS table is given in the paper The existence of HMOLS with equal sized holes by Abel, Bennett and Ge (to appear in Discrete Math). The authors were kind enough to send the pdf file for the new table. Click here for this table. R. Julian R. Abel (julian@maths.unsw.edu.au). April 2001.

Page 173, Theorems 4.15, and 4.16: The following TDs exist: TD2(8,22) and TDλ(9,21) for any λ > 1. R. Julian R. Abel (julian@maths.unsw.edu.au). Feb. 2000.

Page 173, Theorem 4.16: J. Abel, I. Bluskov and M. Greig have the following new results for TDs with index > 1:
(1) A TD3(10,n) exists for n not in {5,6,14,20,35,45,55,56,60,78,84,85,102}.
(2) A TDλ(8,n) with λ>1 exists for all n, λ except for λ = 2, v = 2, 3 and possibly for λ = 2, n = 6, 34.
(3)A TD9(10,n) exists for all n except possibly 35.
(4) A TDλ(9,n) with λ > 1 exists if n, λ are not given below:

• λ = 2 and n is in {2,3,6,14,34,38,39,50,51,54,62}.
• λ = 3 and n= 5, 45 or 60.
• λ = 5 and n= 6 or 14.
R. Julian R. Abel (julian@maths.unsw.edu.au). Sept. 2002.