# New Results in Part III

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 III of the Handbook (pages 183-226).

Last updated 10/30/03

Page 191, Theorem 1.34. There exists a 4-GDD of type 6um1 for every u &ge 4 and m &equiv 0 (mod 3) with 0 &le m &le 3u-3 except for (u,m)=(4,0) and except possibly for (u,m) \in {(7,15),(11,21),(11,24), (11,27), (13,27),(13,33), (17,39),(17,42),(19,45), (19,48),(19,51), (23,60),(23,63)}. Many small designs were needed in this construction, they can be found here. This is a result of Ge Gennian and Rolf Rees (to appear in Discrete Mathematics). (gnge@emba.uvm.edu), Oct. 2003.

Page 191, Theorem 1.34. Further results on 4-GDD's:
1. There exists a 4-GDD of type 2um1 for each u&ge 6 with u &equiv 0 (mod 3), m &equiv 2 (mod 3) and with 2 &le m &le u-1 except for (u,m)=(6,5) and possibly excepting (u,m) \in {(21,17), (33,23), (33,29), (39,35), (57,44)}.
2. There exists a 4-GDD of type 4um1 for each u &ge 6, u &equiv 0 (mod 3) and m &equiv 1 (mod 3) with 1 &le m &le 2(u-1).
3. A 4-GDD of type 5um1 exists if and only if either u &equiv 3(mod 12) and m &equiv 5(mod 6), 5 &le m &le (5u-5)/2; or u &equiv 9(mod 12) and m &equiv 2(mod 6), 2 &le m &le (5u-5)/2; or u &equiv 0(mod 12) and m &equiv 2 (mod 3), 2 &le m &le (5u-8)/2.
4. There exists a 4-GDD of type 12um1 for each u&ge 4 and m &equiv 0 (mod 3) with 0 &le m &le 6(u-1).
5. A 4-GDD of type 15um1 exists if and only if either u &equiv 0(mod 4) and m &equiv 0 (mod 3), 0 &le m &le (15u-18)/2; or u &equiv 1(mod 4) and m &equiv 0(mod 6), 0&le m &le (15u-15)/2; or u &equiv 3(mod 4) and m &equiv 3(mod 6), 0< m &le (15u-15)/2.
All of these results are by Ge and Ling. (gnge@emba.uvm.edu), Oct. 2003

Page 192, Table 1.38. A (115,{4,5,11*},1) PBD may be formed from a (104,{4,5},1) PBD with 21 parallel classes of 4-blocks, given by Ling on p. 57 of his PhD thesis (Waterloo 1997) so adding 11 points gives a (115,{4,5,11*},1) PBD. (greig@sfu.ca) Mar. 2000.

Page 192, Table 1.38. Now the only possible exceptions for (v, {5} U k*) PBDs with k in {17,21,25,29} are for (v,k) = (77,17), (89,17), (137,17), (141,25), (137,29) and (397,29). R.J.R. Abel and A.C.H. Ling, Some New Constructions for (v,{5,w*},1) PBDs, JCMCC 32 (2000), 97-102. (julian@alpha.maths.unsw.edu.au). Sept. 1997.

Page 197, Theorem 2.30. This theorem has been generalized to the following: If q is a prime power, then there exists a {h, q + h}-GDD of type (q^2 - q)^{q+1}(q^2)^h, in which each block intersects any group of size q^2, where 1 &le h &le q^2 - q. (See Y. Miao and L. Zhu, On resolvable BIBDs with block size five, Ars Combin. 39 (1995), 261 - 275). Ying Miao (miao@sk.tsukuba.ac.jp).

Page 197, Corollary 2.34. If q is a prime power, then there exists a {q^2}-GDD of type (q^2-q)^{q^2+q+1}. (i.e. There are no blocks of size q^2-q in this construction). M. Greig (greig@sfu.ca) Nov. 1997.

Page 206, Table 3.17. The closure of B(4,5,6) is now known to contain 47 (this appears in the final version of ref. [21] in the section).

Page 206, Table 3.17. This table has been extended by Martin Gruttmuller to contain the closure of all subsets of {3,4,...,22} containing 3 and the closure of all sets {3,k} with 4 &le k &le 50, all with no possible exceptions. These new results can be found at ftp://www.math.uni-rostock.de/pub/members/mgruttm/closure.htm. (mgruttm@zeus.math.uni-rostock.de) Sept. 1997.

Page 206, Table 3.17. Ron Mullen and Martin Gruttmuller extended this table to contain the closure of all sets {4,k} with 5 &le k &le 15. These new results can be found here . (mgruttm@zeus.math.uni-rostock.de) Nov. 1998.

Page 206, Table 3.17: 44, 45, 47 are definite exceptions for B(4,7,8). Malcolm Greig (greig@sfu.ca). Jan 1998. Click here for the construction.

Page 207, Table 3.17. 48 is a definite exception for B(4,7,9). Malcolm Greig (greig@sfu.ca) Jan 1998. Click here for the construction.

Page 207, Table 3.17. 74 in B(6,8,9) (so in B(4,6,8,9)). (Remove a fano subplane and an external line from Hughes(9) plane). M. Greig (greig@sfu.ca) Nov. 1997.

Page 207, Table 3.17. 259 is in B(5,7). R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). Sept. 1997.

Page 207, Table 3.17: The PBD closure for K={4, 9, 12} is v=0, 1, 4, 9 (mod 12) with the definite exception of v=21, 24. This follows from the K={4, 9} result, the existence of 4-GDDs of type 12^n for all n > 3 and, for v=69, 93, by truncating the largest group of a 5-GDD of type 4^6 or 4^7 8^1 to size 3, then giving all points a weight of 3 in Wilson's construction. Malcolm Greig (greig@sfu.ca) June 2003.

Page 207 -- 212 , Table 3.17: Click here for a .pdf file containing new results on PBDs with minimum block size 5 and also a table for B(Q_{\geq 8}). This is joint work of Julian Abel and Frank Bennett. R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). Sept 2003.

Page 208, Table 3.17. 49 in B(5,6,8). Bierbrauer [The maximal size of a 3-arc in PG(2,8), (preprint)] exhibits a pentario, or pio, consisting of three quintuples, any pair of which form a hyperoval. Its complement is a 58 point {6,7,9}-arc. Removing a 9-line gives the PBD. (greig@sfu.ca) April 1998.

Page 209, Table 3.17. {77, 170, 174, 175, 176} are in B(6,7,8) and B(6,7,8,9). R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). Sept. 1997 and April 2000.

Page 209, Table 3.17. The Janko/Tonchev (175,7,1) BIBD also gives 176 in B(7,8), and similarly, Abel's (259,7,1) BIBD also gives 260 in B(7,8), so 175, 176, 259, and 260 are not exceptions for B(7,8,9). M. Greig (greig@sfu.ca) Nov. 1997. Furthermore, Colbourn, Lamken, Ling, and Mills have established that the known (175,7,1) design has exactly 501 parallel classes, and using the structure of these have established that 177, 178, 179 are not exceptions for B(7,8,9). C. Colbourn (Charles.Colbourn@uvm.edu) Apr 2000.

Page 210, Table 3.17, Q_{>=8}. The set {1180, 1196, 1198, 1212, 1214, 1236, 1238, 1246, 1254, 1270} is contained in B(Q_{>=8}). If v = 2 (mod 8), then v is in B(Q_{>=8}) for all v >= 1706, except possibly for 1714, 1722, 1730, 1738, 1754, 1762, 1786, 1802. Details for both of these results are available from Frank Bennett, (FBennett@linden.MSVU.ca). Nov. 1999.

Page 211, Q(1 mod 5) U {6}. The values 231, 261, 266, 276, 296, 376, 741, 801 and 946 can be deleted from the list of exceptions. R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). May 1997.

Page 211, Q(0,1 mod 7). The values 259 and 260 can be deleted from the list of exceptions. R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). Sept. 1997. The values 175 and 176 can be deleted from the list of exceptions. M. Greig (greig@sfu.ca). Dec. 1997.

Page 211 Table 3.19 A generating set for `H_{0,1(4)}\setminus\{4,5\}$is: 8,9,12,13,16,17,20,21,24,25,28,29,32,33,36,37,40,41,44,45,48,49,52,53, 56, 60,61, 68,69, 76,77, 84,85,88, 92,93, 101. All these elements are essential with the possible exception of 101. M. Greig (greig@sfu.ca) Sept/Dec. 1999. Page 212, H1(5). The following values don't need to be included in the generating set for H1(5): 131, 146, 166, 196, 221, 226, 231, 251, 261, 266, 296, 316, 326, 351, 356 (Abel, Greig) R. Julian R. Abel (julian@alpha.maths.unsw.edu.au). May 1997. Page 212, H_{0,1 mod 6}. Essential elements: 6 7 12 13 18 19 24 25 30 36 37. Possibly needed: 54 55 60 61 102. (Note: 97 in B(6,7) since a 6-GDD of type$6^16$is known). B. Du, On finite bases for some PBD-closed sets, JCMCC 20 (1996) 217-224. Reported by M. Greig (greig@sfu.ca) Nov. 1997. Page 212, H_{0,1 mod 7} Essential elements: 7 8 14 15 21 22 28 29 35 36 42 43. Possibly needed: 70 71 77 78 84 85 106 126 127 133 134 140 141 154 155 161 162 168 182 189 238 239 253 266 267 273 274. (Note: 175,259,260 in B(7,8) since 7-GDDs of type$7^25$and$7^37\$ are known, and 147,148,245,246,252 are removed by TD(7,n) for n=21,35,36). B. Du, On finite bases for some PBD-closed sets, JCMCC 20 (1996) 217-224. Reported by M. Greig (greig@sfu.ca) Nov. 1997.

Page 212, H_{0,1 mod 8} Essential elements: 8 9 16 17 24 25 32 33 40 41 48 49 56. Possibly needed: 88 89 96 97 104 105 112 113 160 161 168 169 176 177 184. B. Du, On finite bases for some PBD-closed sets, JCMCC 20 (1996) 217-224. Abel, Bluskov and Greig have now shown 88 is essential. M. Greig (greig@sfu.ca) Dec. 1999.

Page 206-213, Tables 3.17--3.19, References. All the updates to the tabled sets known to me as of Dec. 1999 were recorded in the paper, "Some Pairwise Balanced Designs", which is essentially [12] on p. 213. This paper is published in The Electronic Journal of Combinatorics vol 7(1) (2000) R13 and is available from http://www.combinatorics.org/Volume_7/Abstracts/v7i1r13.html. (greig@sfu.ca) Mar. 2000.

Page 213, Reference [19]. This paper published in JCTA 77(1997), 228-245. M. Greig (greig@sfu.ca). Jan. 1999.

Page 216, Table 4.19. LIN(12) = 28872973. Anton Betten (anton@btm2xf.mat.uni-bayreuth.de). Sept. 1997.

Page 216, Table 4.19. PLIN(17) = 161913. Dieter Betten (betten@btm2xf.mat.uni-bayreuth.de).

Page 216, Table 4.19. PLIN(18) = 2412890. A. Betten and D. Betten, Note on the proper linear spaces on 18 points, Algebraic Combinatorics and its Applications (ed. Betten, Kohnert, Laue, Wasserman), Springer, 2001. pp 40-54.

Page 225, Theorem 6.14. This theorem can be updated as follows:
There exists a (4,1)-frame of type h^{u} if and only if u > 4, h = 0 mod 3, and h(u-1) = 0 mod 4, except possibly where
1. h = 36 and u \in {8, 12};
2. h = 24 and u = 12;
3. h = 6 mod 12, h \ne 18 and u \in {7, 19, 23, 27, 35, 39, 43, 47, 63, 67};
4. h=18 and u \in {15, 17, 19, 23, 27, 39}.
Ge Gennian (gnge@public1.sz.js.cn). October 2000.