# New Results in Part III

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 III of the Handbook.

Last edited 1/19/17

Page 149, Remark 1.102. There is no Sudoku critical set of size 16. The reference is: Gary McGuire, Bastian Tugemann, Gilles Civario, There is no 16-Clue Sudoku: Solving the Sudoku Minimum Number of Clues Problem, arXiv:1201.0749v1 (January 2012). Gordon Royle has collected 49,151 distinct Sudoku critical sets of size 17. These can be accessed on his website http://mapleta.maths.uwa.edu.au/~gordon/sudokumin.php

Page 175, Table 3.83 and Page 176 Table 3.87. In Table 3.83, i7 can be reduced to 618. Here's how: For the 3 largest unknown cases, write
702 = 23.29 + 18 (spike) + 17
660 = 23.27 + 14 (spike) + 25
654 = 23.27 + 14 (spike) + 19
So there are 7 HMOLS of types 2328 411 171 , 2326 371 251 and 2326 371 191 . Filling the holes with 0 extra points respectively for v= 702, 660, 654 one obtains 7 idempotent MOLS for these values. You can also get 660 from 7 HMOLS(878 351 ) (660 = 8.79 + 27 + 1). Julian Abel , June 2014.

Page 176 Table 3.87. N(7968) >= 31 (not 30). In 1969 Denniston showed there exists a resolvable (v,32,1) BIBD for v = 7968 = 213 - 28 + 25. This RBIBD gives the required MOLS. Julian Abel , June 2014.

Page 190, Table 3.102. This table should have included 4 maxMOLS(9), since they appeared in reference [750]. Also, there exists 2 maxMOLS(n) for all n>6 that aren't twice a prime > 7. So "2" should be added as a value of k in Table 3.102 for all n > 6 except 22, 26, 34, 38, 58. The reference is P.Danziger, I.M.Wanless and B.S.Webb, "Monogamous Latin Squares", J. Combin. Theory Ser. A 118 (2011), 796--807. Ian Wanless (ian.wanless@monash.edu) Jan 2011.

Page 214, Theorem 5.23.  An ISOLS(26,8) has been found.  See Example 2.8 in  Hantao Zhang, 25 new r-self-orthogonal Latin squares, Discrete Math. 313 (2013), 1746–1753.