# New Results in Part 1

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 1 of the Handbook (pages 1-94).

Last edited 5/22/05

Page 15, Design #29. Nd = 11,084,874,829 (instead of Nd $≥$ 1.1x10^9). Reference: P. Kaski and P.R.J. Östergård, "The Steiner triple systems of order 19", submitted for publication. A preprint of the paper can be found at http://www.tcs.hut.fi/~pkaski/sts19.ps. Patric Östergård (patric.ostergard@hut.fi), December 2001.

Page 15, Design #32. Nd4492 (instead of Nd145). Vedran Krcadinac has classified (28,4,1) designs with nontrivial automorphisms. The number of such designs is 4466. The work was submitted to Glasnik matematicki (www.math.hr/~glasnik). Brouwer found 26 designs with trivial automorphism groups (page 36, [17]), so there are at least 4492 non-isomorphic (28,4,1) designs. Vedran Krcadinac (krcko@cromath.math.hr), February 2001.

Page 15, Design #34. Nd = 0 (instead of ?). Reference: S.K. Houghten, L.H. Thiel, J. Janssen, and C.W.H. Lam, "There is no (46,6,1) block design", J. of Combinatorial Designs 9 (2001), pp 60 - 71.

Page 15, Design #42. Nd 2156186 (instead of Nd 2 x10^6) this is the number of STS(21) which have three STS(7) subsystems; Nr 13036 (instead of Nr 192), Reference: P. Kaski, P. R. J. Ostergrd , S. Topalova, and R. Zlatarski, Steiner triple systems of order 19 and 21 with subsystems of order 7, Discrete Mathematics, to appear. Svetlana Topalova (svetlana@moi. math.bas.bg), November 2004.

Page 15, Design #45. Nd15 (instead of Nd5). Vedran Krcadinac has classified (41,5,1) designs with automorphisms of order 3 and has found a (41,5,1) design with a single involution. This work will appear in JCMCC. Vedran Krcadinac (krcko@cromath.math.hr), February 2001.

Page 15, Design #46. Nd >= 22998 (Instead of Nd 35). Reference: S. Topalova "Enumeration of 2-(21,5,2) designs with automorphisms of an odd prime order", Diskretnii Analiz i Issledovanie Operatsii, vol.5, N1 (1998), 64-81 (in Russian). Svetlana Topalova (svetlana@moi.math.bas.bg), August 1998.

Page 15, Design #55. Nd = 242,995,846, and Nr = 74,700. Reference: P. R. J. Östergård, "Enumeration of 2-(12,3,2) designs", Australas. J. Combin. 22 (2000), 227-231. (Patric.Ostergard@hut.fi), May 2001.

Page 15, Design #56, Nr=5 (instead of Nr 1). Refencence: LB Morales and C Velarde, A complete enumeration of (12,4,3)-RBIBDs (submitted to JCD), Luis B. Morales (lbm@servidor.unam.mx), September 2000.

Page 15, Design #66. Nr = 426. P. R. J. Östergård and P. Kaski, Enumeration of 2-(9,3,lambda) designs and their resolutions, submitted for publication. Patric Östergård (Patric.Ostergard@hut.fi), April 2000.

Page 15, Design #72. Nd 118884 (instead of Nd 28), and Nr 748 (instead of Nr 28). Reference: S. Topalova (lpmivt@bgcict.acad.bg), "Enumeration of 2-(25,5,2) Designs with an automorphism of order 5", J. Combinatorics, Information and System Sciences, No.3-4, 161-176 (1997). Svetlana Topalova (svetlana@moi.math.bas.bg), August 1998.

Page 15, Design #75. Nd 236 (Instead of Nd 1). Reference: S.N. Kapralov and S. Topalova, "Enumeration of 2-(21,6,3) Designs with Automorphisms of Order 7 or 5", Ars Combinatoria 48 (1998), 135-146. Svetlana Topalova (svetlana@moi.math.bas.bg), August 1998.

Page 16, Design #87. Nr 2. In fact, there is a 1-rotational (40,4,1)-RBIBD over Z_39. This design cannot be isomorphic to the point-line design of PG(3,3) because the full stabilizer of a point in PG(3,3) does not admit Z_39 as a subgroup. M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 16, Design #97. Nd 1,108,800 (instead of Nd 389). Lam, C. W. H., Lam, S., and Tonchev, V. D., Bounds on the number of Affine, Symmetric and Hadamard Designs and Matrices'', J. Combinatorial Theory, Series A (to appear). C. Lam (lam@cs.concordia.ca), Jan. 2000.

Page 16, Design #99. Nr 36 (instead of Nr 21). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 16, Design #102. Nd 896 (instead of Nd 207). D.Tiessen and G.H.J. Van Rees, "Many (22,44,14,7,2) - and (15,42,14,5,4) balanced incomplete block designs", Journal of Statistical Planning and Inference, 62(1997), 115-124. Ben Li (lipakc@cs.umanitoba.ca), March 2003.

Page 16, Design #102. Nr = 0. (instead of ?). Reference: P. Kaski and P. R. J. Östergård, "There exists no (15,5,4) RBIBD", J. Combin. Des. 9 (2001), 227-232. (Patric.Ostergard@hut.fi), May 2001.

Page 16, Design #108. Nd 3393 (instead of Nd 34). D.Tiessen and G.H.J. Van Rees, "Many (22,44,14,7,2) - and (15,42,14,5,4) balanced incomplete block designs", Journal of Statistical Planning and Inference, 62 (1997), 115-124. Ben Li (lipakc@cs.umanitoba.ca), March 2003.

Page 16, Design #123. Nd 294 (instead of Nd 11). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 16, Design #124 Nd 76 (instead of Nd 30). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 16, Design #128. Nd 25 (instead of Nd 15). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 17, Design #140. Nd 72 (Instead of Nd 8). Sanja Rukavina proved there are exactly 72 symmetric (71,15,3) designs admitting an action of cyclic group of order 6 acting with one fixed point. Sanja Rukavina (sanjar@mapef.pefri.hr), June, 1999.

Page 17, Design #143. Nd 11,727,788 (instead of Nd 1,266,891). Lam, C. W. H., Lam, S., and Tonchev, V. D., Bounds on the number of Affine, Symmetric and Hadamard Designs and Matrices'', J. Combinatorial Theory, Series A (to appear). C. Lam (lam@cs.concordia.ca), Jan. 2000.

Page 17, Design #144. Nr 529 (instead of Nr 28). M.Buratti and F.Zuanni have established that there are exactly 500 1-rotational KTS(33). These new KTS(33) (including the Kageyama RBIBD) must be added to the 28 RBIBD's by Tonchev and Vanstone and the Ball RBIBD. F.Zuanni (zuanni@ing.univaq.it), May 1998.

Page 17, Design #145. Nr = 149,041. P. R. J. Östergård and P. Kaski, Enumeration of 2-(9,3,lambda) designs and their resolutions, submitted for publication. Patric Östergård (Patric.Ostergard@hut.fi), May 2001.

Page 17, Design #146. Nd 769 (instead of Nd 236).
a) 224 are cyclic (Ref [28]);
b) 12 are regular over V49 (Ref [48]);
c) 529 are obtainable by composition of a (33,3,1)-RBIBD with the affine plane of order 16 (Ref: M. Buratti and F. Zuanni, "Composing a (r(k-1)+1,k,1)-RBIBD with a (r,k+1,1)-BIBD", preprint).
d) 4 are 1-rotational over the group G = (Z3)+(Z4)+(Z4) (Ref: M. Buratti and F. Zuanni, "On singular 1-rotational difference families"). The following blocks are representatives of the block-orbits under G of one of these 1-rotational BIBD's: {000, 100, 200, *}, {000, 010, 020, 030}, {000, 001, 002, 003}, {000, 011, 022, 033}, {000, 012, 102, 231}, {000, 013, 120, 221}, {000, 021, 103, 213}.
M. Buratti (buratti@mat.uniroma1.it), Dec. 1997. (Further report by F. Zuanni (zuanni@ing.univaq.it), Feb. 1998.)

Page 17, Design #148. Nd 542 (instead of Nd 1). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 17, Design #150. Nd = 270,474,142. P. R. J. Östergård, There are 270,474,142 nonisomorphic 2-(9,4,6) designs, Journal of Combinatorial Mathematics and Combinatorial Computing, 37 (2001), pp 173-176. Patric Östergård (Patric.Ostergard@hut.fi), April 2000.

Page 17, Design #158. Nd 28 (instead of Nd 11). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 17, Design #170. Nd 6 (instead of Nd 1). Reference: I.Landjev, S.Topalova, New symmetric (61,16,4) designs invariant under the dihedral group of order 10, Serdika Math. J., 24 (1998), 179-186. June 1997.

Page 17, Design #173. Nr 173 (instead of Nr 1). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 17, Design #174. Nr 30 (instead of Nr 1). In fact: There are exactly six non-isomorphic cyclically resolvable cyclic (52,4,1) designs (found by C. Lam, Y. Miao and M. Mishima). There are exactly two non-isomorphic (52,4,1)-RBIBDs which are regular over Z_2+Z_2+Z_13 (found by R.J.R. Abel). Finally, there are exactly 22 non-isomorphic cyclically resolvable 1-rotational (52,4,1)-RBIBDs (found by M. Buratti and F. Zuanni). F.Zuanni (zuanni@ing.univaq.it), May 1998.

Page 17, Design #174. Nr 7 (instead of Nr 1). Six new non-isomorphic cyclically resolvable cyclic (52,4,1) designs have been found. Together with the cyclically resolvable 1-rotational (52,4,1) design, there are now at least 7 non-isomorphic resolvable (52,4,1) designs. Ying Miao (miao@sk.tsukuba.ac.jp). July 1997.

Page 17, Design #176. Nd 582 (instead of Nd 3). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 17, Design #185. Nd 4 (Instead of Nd 1). There are 4 designs with automorphisms of order 13. Svetlana Topalova (svetlana@moi.math.bas.bg), August 1998. Further Update: Nd 61. There are 59 designs admitting an action of cyclic group of order 6. Between them, there are 2 designs with automorphisms of order 13. Sanja Rukavina (sanjar@mapef.pefri.hr), August 1999.

Page 18, Design #199. Nd 1535 (instead of Nd 1). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 18, Design #205. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". Journal of Combinatorial Designs 8 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 18, Design #208. Nd 108 (instead of Nd 7). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 18, Designs #219. Nr 380 (instead of 88). There are exactly 292 non-isomorphic cyclically resolvable cyclic Steiner triple systems of order 39. Together with the known 88 non-isomorphic Steiner triple systems of order 39, which are in fact cyclically resolvable 1-rotational, there are now at least 381 non-isomorphic Kirkman triple systems of order 39. (See C. Lam and Y. Miao, Cyclically resolvable cyclic Steiner triple systems of order 21 and 39, preprint, 1998). Ying Miao (miao@sk.tsukuba.ac.jp). March 1998.

Page 18, Design #220. Nd 10040 (instead of Nd 1). Nr 204 (instead of Nr 3). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 18, Design #221. Nd 10067 (instead of Nd 1). Nr 14 (instead of Nr 2). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 18, Design #233. Nd 5.87 x 1014 (instead of Nd 38). Lam, C. W. H., Lam, S., and Tonchev, V. D., Bounds on the number of Affine, Symmetric and Hadamard Designs and Matrices'', J. Combinatorial Theory, Series A (to appear). C. Lam (lam@cs.concordia.ca), Jan. 2000.

Page 18, Design #235. Nd = 5,862,121,434 and Nr = 203,047,732. P. R. J. Östergård and P. Kaski, Enumeration of 2-(9,3,\lambda) designs and their resolutions, submitted for publication. Patric Östergård (Patric.Ostergard@hut.fi), May 2001.

Page 18, Design #242. Nd 2 (instead of Nd 1). Reference: M.Weidenfeld, Construction of two nonisomorphic (11,55,20,4,6) balanced incomplete block designs having the same automorphism group, Europ. J. Combin. 13(1992), 515-520. June 1997. Nd 348. P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 18, Design #246. Nd 7260 (instead of Nd 1). P. Dobcsányi and L.H. Soicher, An on-line collection of t-designs, http://designtheory.org/database/t-designs/ , May 2005.

Page 19, Design #255. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". J. Comb. Des. 8 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 19, Design #256. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". J. Comb. Des. 8 (2000), 261-2738 (2000), 261-2738 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 19, Design #279. Nd 11 (instead of Nd 10). The ten (85,5,1)-BIBD's of the Handbook are cyclic (Ref [28]) but the (85,5,1)-BIBD obtainable composing a (64,4,1)-RBIBD with the projective plane of order 4 cannot be cyclic (Ref: Same as for designs #146c above). M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 19, Design #291. Nd 1 (instead of Nd = ?). Julian Abel (julian@maths.unsw.edu.au), Sept. 2000

Page 19, Design #300. Nd 10,810,800 (instead of Nd 157). Lam, C. W. H., Lam, S., and Tonchev, V. D., Bounds on the number of Affine, Symmetric and Hadamard Designs and Matrices'', J. Combinatorial Theory, Series A (to appear). C. Lam (lam@cs.concordia.ca), Jan. 2000.

Page 20, Design #312. Nd 82 (Instead of Nd 2). Reference: Svetlana Topalova, Classification of Hadamard matrices of order 44 with Automorphisms of order 7, Discrete Mathematics, Vol. 260 (2003), 275-283. Svetlana Topalova (svetlana@moi.math.bas.bg). August 1998.

Page 20, Design #326. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". J. Comb. Des. 8 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 20, Design #351, Nd 55 (instead of Nd 1). Up to isomorphism there are exactly 54 symmetric (47,23,11) designs admitting an automorphism group isomorphic to a Frobenius group of order 55. Those designs are not isomorphic to the (47,23,11) design which has been constructed via a cyclic difference set. Dean Crnkovic (deanc@mapef.pefri.hr) Jan. 2001.

Page 21, Design #376. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". J. Comb. Des. 8 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 21, Design #407. Nd 28. (Instead of Nd 7). Up to isomorphism there are 22 symmetric (70,24,8) designs with automorphism groups isomorphic to Frob21 x Z2. Among them there are four self-dual and nine pairs of dual designs. Full automorphism groups of those designs are isomorphic to Frob21 x Z2. One of self-dual designs is isomorphic to the design constructed by Z. Janko and Tran van Trung the other designs are new. Dean Crnkovic (deanc@pefri.hr) Sept. 1999.

Page 21, Design #410. Nd 169,574 (instead of Nd 32). (Ref: Same as for designs #146c above). M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 21, Design #412. Nd 3. The (101,5,1)-BIBD of the CRC Handboock is cyclic. To construct a noncyclic one combine a (76,4,1)-RBIBD, with a (25,5,1)-BIBD, say D. In this way we obtain a (101,5,1)-BIBD, say E, admitting D as unique (25,5,1) subdesign. Of course E cannot be cyclic (any translate of a subdesign D of a cyclic design E also is a subdesign of E) and hence not isomorphic to the cyclic (101,5,1)-BIBD of the Handbook. M. Buratti, Sept. 1997.
A new design is 1-rotational over the group G = (Z2)+(Z2)+(Z5)+(Z5). (Ref: Same as for designs #146c above). The following blocks are representatives for the block-orbits under G of this design: {0000,1000,0100,1100,*},{0000,0011,0022,0033,0044},{0000,0012,0024,0031,0043}, {0000,0013,0021,0034,0042},{0000,0014,0023,0032,0041}, {0000,0010,1014,0104,1133}, {0000,0004,1024,0120,1142}, {0000,0020,1023,0103,1111}, {0000,0003,1043,0140,1134}. M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 22, Design #448. Nr 1. (Abel, Greig Australasian JC 15(1997), 177-202)

Page 22, Design #451. Nr = 1,363,486 (instead Nr 1). It is also noted that 1,360,800 of which have trivial full automorphism group. Reference: Kaski, Petteri; Morales, Luis B.; Ostergerd, Patric R. J.; Rosenblueth,David A.;Velarde, Carlos, Classification of resolvable 2-(14,7,12) and 3-(14,7,5) designs. J. Combin. Math. Combin. Comput. 47 (2003), 65--74.

Page 22, Design #452. Nd 1. Abel, Bluskov and Greig have found an example of this BIBD. A description of it can be found here . (greig@sfu.ca) Nov. 1999.

Page 23, Design #513. Nd 1463. Reference: D. Held, M.-O. Pavcevic, Symmetric (79,27,9)-designs admitting a faithful action of a Frobenius group of order 39, Europ. J. Combin. 18(1997), 409-416. June 1997. - The full automorphism groups of these 1463 designs all are direct products of Frob39 by a subgroup of D8. The case Z4 x Frob39 does not occur. The distribution of the designs by the orders of their full automorphism groups is as follows: (411,39), (668,78), (312,156), (72,312). Mario Pavcevic (mario@zpm.etf.hr). Corrected Oct. 1998.

Page 23, Design #526. Nd 1 (instead of ?). W.H. Mills (bill@ccr-p.ida.org). June 1997.

Page 25, Design #574. This design (with parameters 2-(175,7,1)) has been found by Janko and Tonchev. Contracting that design gives a generalized Bhaskar-Rao design GBRD(25,100, 28,7,1,Z7) (See page 241 in the Handbook). (tonchev@math.mtu.edu). July 1997.

Page 25, Design #624. Nd 2 (instead of 1). A (55,10,5) design with no repeated blocks can be constructed from the three base blocks {0 3 6 12 13 20 22 24 35 49}, {5 16 21 26 29 30 34 45 51 53}, {11 18 19 23 33 36 38 39 40 50} developed in Z_55. Ian Wakeling (ian@qistatistics.co.uk), February 2003.

Page 25, Design #630. Nd 1 (instead of ?). Reference: L.B. Morales "Constructing difference families through an optimization approach: six new BIBDs". J. Comb. Des. 8 (2000), 261-273, Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 26, Design #661. Nd 2 and Nr 2. Besides AG(3,5), there is a 1-rotational (125,5,1)-RBIBD. Click here for the base blocks of this design. M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 26, Design #662. Nd 2. In fact, using the (125,5,1)-RBIBD described above, it is possible to get a (156,6,1) which of course cannot be isomorphic to the point-line design of the projective geometry PG(3,5).

Page 26, Design #682. Nr 31 (instead of Nr 1) (2#144). M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 26, Design #684. $Nd ≥ 5,985$ (instead of $Nd ≥ 2$) M. Buratti (buratti@mat.uniroma1.it), March 2002.

Page 26. Design #685. $Nd ≥ 770$ (instead of $Nd ≥ 237$) (F. Zuanni (zuanni@ing.univaq.it), May 1998.

Page 27, Design #736. Nd 5985 (instead of Nd 2). M. Buratti (buratti@mat.uniroma1.it), Nov 1999.

Page 28. Design #789. $Nr ≥ 1$. R. Julian R. Abel (julian@maths.unsw.edu.au), Nov. 1998.

Page 29, Design #823. Nd 2. In addition to the cyclic (simple) design found by Hanani, there is a 1-rotational (not simple) design whose base blocks are B_1 = {0,8,11,16,17}; B_2 = 9B_1; B_3 = 17B_1; B_4 = {0,13,19,23,25}; B_5 = 9B_4; B_6 = 17B_4; B_7 = {0,7,14,21,$\infty$} (5 times). M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 29, Design #829. A design with parameters (106,7,2) has been shown to exist (This also updates Table 2.8). R. Julian R. Abel (julian@maths.unsw.edu.au), March 1998.

Page 29, Design #837. Nr 6 (instead of ?). Reference: L.B. Morales "Two new 1-rotational (36,9,8) and (40,10,9) RBIBDs", Journal of Combinatorial Mathematics and Combinatorial Computing 36 (2001), 119-126. Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 29, Design #840: Nd 1 (instead of ?). Julian Abel (julian@maths.unsw.edu.au), Sept. 2000

Page 30, Design #873. Nd 9. Besides the design which can be constructed via a cyclic difference set, there are precisely 8 further symmetric designs for (71,35,17) admitting a faithful action of a Frobenius group of order 21 in such a way that an element of order 3 fixes precisely 11 points. Five of these designs have 84 and three have 420 as the order of the full automorphism group G. If |G| = 420, then the structure of G is unique and we have G = (Frob21 x Z5):Z4. In this case Z(G) = <1>, G' has order 35, and G induces an automorphism group of order 6 of Z7. If |G| = 84, then Z(G) is of order 2, and in precisely one case a Sylow 2-subgroup is elementary abelian. Dean Crnkovic and Dieter Held (deanc@mapef.pefri.hr), Dec. 2000.

Page 30, Design #880. Nd 1014 and Nr 105. P. R. J. Östergård and P. Kaski, Enumeration of 2-(9,3,lambda) designs and their resolutions, submitted for publication. Patric Östergård (Patric.Ostergard@hut.fi), May 2001.

Page 30, Design #882. $Nd ≥ 16,987,430,331,445$ (instead of $Nd ≥ 2$). This may be obtained combining the update on design #32 given above by Vedran Krcadinac with a result by Buratti and Zuanni (Disc. Math. 238). M. Buratti (buratti@mat.uniroma1.it), March 2002.

Page 30, Design #889. Nr 1. (Abel, Greig, Australasian J. Comb. 15(1997), 177-202)

Page 30, Design #891. Nr = 27,121,734 (instead of Nr 1). Of these, exactly 2,006,690 are simple. (Morales and Velarde, J. Combinatorial Designs, to appear).

Page 30, Design #896. A cyclic difference family for this design has been added to the web page http://www.emba.uvm.edu/~dinitz/newresults.part4.html under Page 273 -- Example 10.10.

Page 31. Design #972. Nd 16,987,430,331,445 (instead of Nd 2) and Nr 210 (instead of Nr 1). M. Buratti (buratti@mat.uniroma1.it) Nov. 1999 and March 2002

Page 32, Design #1030. Nd = 1033129 (instead of >417). (Mathon, Pietsch). Reference: #90 on page 39 of the Handbook.

Page 32, Design #1033. This design (with parameters (196,6,1)) exists. R. Julian R. Abel (julian@maths.unsw.edu.au), Dec. 1997.

Page 33, Design 1040. Nd 1 (instead of Nd=?). Julian Abel (julian@maths.unsw.edu.au), Sept. 2000.

Page 33, Design #1042. Nr 3 (instead of ?). Reference: L.B. Morales "Two New 1-rotational (36,9,8) and (40,10,9) RBIBDs", Journal of Combinatorial Mathematics and Combinatorial Computing 36 (2001), 119-126. Luis B. Morales (lbm@servidor.unam.mx) March 2000.

Page 33, Design #1074. Nd 2091. Besides the design which can be constructed via a cyclic difference set (IV.12.22, Table, Paley), there are precisely 194 symmetric designs for (79,39,19) admitting a faithful action of a Frobenius group of order 39 such that an element of order 3 fixes more than one point. Among these there are four designs having Z3 x Frob39 as full automorphism group; the additional 190 designs have Frob39 as full automorphism group. Further, there are an additional 1896 designs such that a Frobenius group of order 57 is acting. The distribution of these designs by the orders of their full automorphism groups is as follows: (1635,57), (192,114), (21,171), (39,342), (9,1026). The structures of these groups are determined by their orders: Frob57, (Z19 x Z2)Z3, Z19.Z9, (Z19 x Z2)Z9, (Z19 x S3)Z9; here, Z9 acts always faithfully on Z19. Dieter Held and Mario-Osvin Pavcevic (mario@zpm.etf.hr), Dec. 1998 and Feb. 1999.

Page 33. Design #1079. Nd 1013 (instead of Nd 107). (Ref: Same as for designs #146c above). M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 34, Design #1096. This design (with parameters (201,6,1)) exists. R. Julian R. Abel (julian@maths.unsw.edu.au), Dec. 1997.

Page 34, Design #1102. Nd 1. Abel, Bluskov and Greig have found an example of this BIBD. A description of it can be found here . (greig@sfu.ca) Nov. 1999.

Page 35, Design #1171. This symmetric design (with parameters (105,40,15)) has just been found by Zvonimir Janko. A description of it can be found here . (janko@mathi.uni-heidelberg.DE)

Page 35, Design #1174. Nd 5. Let D be a (124,4,1)-RBIBD and let E be a (41,5,1)-BIBD. Combining D and E we immediately get a (165,5,1)-BIBD, say F. E is the only (41,5,1) subdesign of F. Hence, since we have at least 4 choices for E, we get 4 pairwise nonisomorphic (165,5,1)-BIBD's. M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 37, Reference [44]. This paper published in JCMCC 27(1998), 33-52. Malcolm Greig (greig@sfu.ca) Jan 1999.

Page 42, Table 2.3. When k=7 and lambda = 1, replace 4387 by 2605. R. Julian R. Abel (julian@maths.unsw.edu.au)

Page 42, Theorem 2.4 and Table 2.8: A (253,7,lambda) BIBD exists for lambda = 2 (and hence for all lambda 2 since Hanani solved lambda=3). This completely solves the existence problem for BIBDs with k=7 and lambda > 2. R. Julian R. Abel (julian@maths.unsw.edu.au), March 1998.

Page 42, Table 2.5. 141, 276, 706, 741, 766, can all be deleted; also 1221 and 1251 are the only possible exceptions > 801. (Abel, Greig, Mills) (julian@maths.unsw.edu.au), May 1997. (v,6,1) BIBDs exist for v=196, 201, 1221, 1251, leaving v=801 the largest unknown case. R. Julian R. Abel (julian@maths.unsw.edu.au), Dec. 1997.

Page 42, Table 2.6. 61, 66, 67, 83 can be deleted. R. Julian R. Abel (julian@maths.unsw.edu.au)

Page 42, Table 2.6. 29 can be deleted. This design exists as a consequence of the existence of design no. 574 on page 25. Vladimir Tonchev. (tonchev@math.mtu.edu). July 1997.

Page 42, Table 2.7. 4 can be deleted. (See item above concerning page 25, design no. 574.). Also the value 29 can be deleted as a consequence of the existence of the design with t=4. Vladimir Tonchev. (tonchev@math.mtu.edu). July 1997.

Page 42, Table 2.7. 6 can be deleted (i.e: a (259,7,1) BIBD exists). R. Julian R. Abel (julian@maths.unsw.edu.au) Sept. 1997.

Page 42, Table 2.7. 44 can be deleted (Greig). (julian@maths.unsw.edu.au), May 1997.

Page 43, Table 2.11. The values t=8, 72, 93, 97, 107, 128, 197, 212, 219 can be removed from this table (Greig). The t=8 design is a (577,9,1) difference family found by M. Buratti; this difference family removes 1 exception in Theorem IV.10.53, page 281. (julian@maths.unsw.edu.au), May 1997.

Page 43, Table 2.12. The values t= 20 and t=60 can be removed from this table (M. Buratti and F. Zuanni, G-invariantly resolvable Steiner 2-designs which are 1-rotational over G, preprint.) Both of these designs are 1-rotational. Click here for the base blocks of both of these designs. (buratti@mat.uniroma1.it), Sept. 1997

Page 43, Remark 2.17. t=39, 50, 51 can be deleted when k=6. R. Julian R. Abel (julian@maths.unsw.edu.au)

Page 43, Table 2.14. A (358,8,4) BIBD exists. (greig@sfu.ca) Aug. 1999. (v,8,4) BIBDs exist for v=85, 99, 155, 358, 365 and (v,8,2) BIBDs exist for v=141, 148. (Abel, Bluskov, Greig), November 1999.

Page 43, Table 2.14. A (358,8,4) BIBD exists. (greig@sfu.ca) Aug. 1999. (v,8,4 ) BIBDs exist for v=85, 99, 155, 358, 365 and (v,8,2) BIBDs exist for v=141, 148. (Abel, Bluskov, Greig), November 1999. Abel, Bluskov & Greig have now shown that (274,8,4) and (323,8,4) BIBDs exists, so (v,8,4) BIBDs exist for all v = 1 mod 7 with the possible exception of v=22. (greig@sfu.ca) Jan. 2000.

Page 43. As a consequence of the revision of (v,8,7) NRBs (see the new results for I.6.23), and a grouplet construction of a (385,9,9) BIBD, One gets: If the necessary condition v \equiv 1 (mod 8) holds, then a (v,9,9) BIBD exists, except possibly for v=553. Abel, Finizio, Greig, Lewis. (greig@sfu.ca) Aug. 1999.

Page 43, Table 2.15: A (135,9,4) BIBD exists and (v,9,8) BIBDs exist for v=54, 198, 315, 1278. (Abel, Bluskov, Greig), November 1999.

Page 43, Remark 2.20. Now that 5 MOLS(42) are known (See Theorem II.2.54), there exists a 7-GDD of type 42t for t= 7, in addition to the known t=8,9. Hence if t is in B({7,8,9}), there is a a 7-GDD(42t). From the table on P209, this means there is a 7-GDD(42t) for t=58 and for 63 <= t <= 92. R. Julian R. Abel (julian@maths.unsw.edu.au)

Page 50, Table 3.31. There exists an infinite family of 3-(q+1, (q+1)/2, (q+1)(q-3)/8)designs, where q is a prime power congruent to 3(mod 4). Reference: S. Iwasaki, An elementary and unified approach to the Mathieu-Witt systems, J. Math.Soc.Japan 40(1988),393-414. S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 54, Table 3.37. There does not exist a 3-(22,10,6) design (and also there is no 3-(37,16,8) design). This was proved in the Ph.D. thesis of J"orn Bolick at Hamburg University. Reported by H.-D. Gronau (gronau@zeus.math.uni-rostock.de). Dec. 19, 1997.

Page 56, Table 3.37. There exists a 3-(24, 12, 12.5) design. This results when q=23 in the theorem by S. Iwasaki mentioned above (Page 50, Table 3.31). S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 56, Table 3.37. There exists a 3-(26, 9, 3.21) design. Reference: S. Iwasaki and T. Meixner A remark on the action of PGL(2, q) and PSL(2, q) on the projective line, Hokkaido Math.J. 26 (1997), 203-209. S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 57, Table 3.37. There exists a 3-(26, 13, 13.11) design, a 3-(28, 14, 28.6) and a 3-(30, 5, 5.3) design. Reference: S. Iwasaki and T. Meixner A remark on the action of PGL(2, q) and PSL(2, q) on the projective line, Hokkaido Math.J. 26 (1997), 203-209. S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 57, Table 3.37. There exists a 3-(28, 14, 14.6) design. This results when q=27 in the theorem by S. Iwasaki mentioned above (Page 50, Table 3.31). S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 57, Table 3.37. There exists a 3-(30, 8, 4.6) design and a 3-(30,8, 8.6) design. Reference: S. Iwasaki, Infinite families of 2- and 3-designs with parameters ... , JCD 5(1997), 95-110. S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 58, Table 3.37. There exists a 3-(30, 15, 15.13) design. Reference: S. Iwasaki, Infinite families of 2- and 3-designs with parameters ... , JCD 5(1997), 95-110. S. Iwasaki (iwasaki@math.hit-u.ac.jp). April 2002.

Page 59-60, Table 3.37. There exist simple 5-(24,10,lambda) designs for lambda=18m, 6<= m <=52 and m = 4. These designs all admit PSL(2,23). These designs are 4-(24,10,60m) designs with 6<= m <=52 and m = 4. In addition, there exist 4-(24,10,60m) designs for m=3, 5, 53, 54, 55. Masaaki Kitazume and Akihiro Munemasa (munemasa@math.kyushu-u.ac.jp). May 28, 1997.

Page 60, Table 3.37. A simple 5-(24,10,36) design has been found by T.A. Gulliver and Masaaki Harada. (harada@math.okayama-u.ac.jp)

Page 62, Remark 3.38. New simple 7-designs and 8-designs have been found by Betten, Laue and Wassermann. Click here for their web page. It contains a wealth of valuable information concerning these designs and the software package DISCRETA which was used to find them. In particular, Alfred Wassermann has just found some very interesting new ones: 7-(20,10,lambda), lambda = 116,124,134. Also found are 8-(31,10,93) and 8-(31,10,100), both with automorphism group PSL(3,5).

Page 63, Table 3.44. There exists a large set LS_360(3,10,24) with automorphism group PSL(2,23). Note there are 323 3-(24,10,360) designs comprising this large set. Anton Betten (anton@btm2xf.mat.uni-bayreuth.de).

Page 79, Known Symmetric Designs. Let q and r = (q^d - 1)/(q - 1) be prime powers. For any positive integer m, there exists a symmetric (v,k,lambda)-design with v = qr (r^{m+1} - 1)/(r - 1) + 1, k = r^{m+1}, lambda = r^m (r - 1)/q. If d = 2, these are precisely the parameters of Family 10. Designs with q = 2 are contained in Family 11. The case q = 8, d = 3 was reported by Yury Ionin earlier. Otherwise, the parameters are new. Yury Ionin (yury.ionin@cmich.edu). Nov. 1998.

Page 79, Known Symmetric Designs. In Family 12, "p a prime" can be replaced by "p a prime power". Yury Ionin (yury.ionin@cmich.edu). Feb. 1997.

Page 79, Known Symmetric Designs. A new infinite class of symmetric designs has the following parameters: v = (73^(m+1) - 64)/9, k = 73^m, lambda = 9x73^(m-1), n = 64x73^(m-1) (m is a positive integer). Yury Ionin (yury.ionin@cmich.edu).

Page 79, Known Symmetric Designs. A new infinite class of symmetric designs has the following parameters: v = (2(q+1)^(2m+1) - q)/(q+2), k = q((q+1)^(2m) - 1)/(q+2), lambda = (q^2(q+1)^(2m-1) - 2q)/(2q+4), n = q(q+1)^(2m-1)/2, where q = 2^p - 1 is a Mersenne prime, m is a positive integer. (For the complementary design, k = (q+1)^(2m), lambda = (q+2)(q+1)^(2m-1)/2). If q = 3, then it is the family claimed by J.D.Fanning in the note "A family of symmetric designs", Discrete Math. 146 (1995), 307-312. Also note that in the case m=1, this family corresponds to a subcase for m=2 of the Family 11 (Wilson and Brouwer) in the Handbook. Yury Ionin (yury.ionin@cmich.edu).

Page 79, Known Symmetric Designs. If m and d are positive integers, p is an arbitrary prime power, and q is a prime power that is specified below, then there exist symmetric designs with the following parameters:

a) v = p^d(q^(2m)-1)/((p-1)(p^d+1)), k = p^d(p^dq^(2m-1)-1)/((p-1)(p^d+1)), lambda = p^d(p^(2d)q^(2m-2)-1)/((p-1)(p^d+1)), q = p^(d+1)+p-1;

b) v = 2.3^d(q^(2m)-1)/(3^d+1), k = 3^dq^(2m-1), lambda = 3^d(3^d+1)q^(2m-2)/2, q = (3^(d+1)+1)/2;

c) v = 3^d(q^(2m)-1)/(2(3^d-1)), k = 3^d(3^dq^(2m-1)-1)/(2(3^d-1)), lambda = 3^d(3^(2d)q^(2m-2)-1)/(2(3^d-1)), q = 3^(d+1)-2.

All these designs are new if m > 1. If m = 1, then design a) is in Family 12, designs b) and c) are in Family 9. Yury Ionin (yury.ionin@cmich.edu). Feb. 1997.

Page 79, Known Symmetric Designs. v = 2^(2d+3)(q^(2m)-1)/(q+1), k = 2^(2d+1)q^(2m-1), lambda = 2^(2d-1)(q+1)q^(2m-2), where d and m are positive integers and q = (2^(2d+3)+1)/3 is a prime power. For m = 1, these designs belong to the Davis-Jedwab family of difference sets (q does not have to be a prime power there); for m > 1, they are new. Yury Ionin (yury.ionin@cmich.edu). Nov. 1997.

Page 79, Known Symmetric Designs. v = 2^(2d+3)(q^(2m)-1)/(3q-3), k = 2^(2d+1)q^(2m-1), lambda = 2^(2d-1)(3q-3)q^(2m-2), where d and m are positive integers and q = 2^(2d+3)-3 is a prime power. For m = 1, these design are the complements of the designs from the previous family; for m > 1, they are new. Yury Ionin (yury.ionin@cmich.edu). Nov. 1997.

Page 79, Known Symmetric Designs. v = h((2h-1)^{2m}-1)/(h-1), k = h(2h-1)^{2m-1}, lambda = h(h-1)(2h-1)^(2m-2), where m is a positive integer, h = 3.2^d or -3.2^d (d is a nonnegative integer), and |2h-1| is a prime power. For m = 1, these designs belong to the Menon family; for d = 0, they belong to the family that is listed above on this web page (p. 79(b,c)). If m > 1 and d > 0, they are new. Yury Ionin (yury.ionin@cmich.edu). Nov. 1997.

Page 79, Known Symmetric Designs. There is a new family of symmetric designs that was obtained by the combined efforts of Kharaghani (2 papers), Janko, Karaghani and Tonchev (two papers), Ionin and Kharaghani, and Ionin. All references can be found in the paper by Yury Ionin titled "Regular Hadamard matrices generating infinite families of symmetric designs". This paper will appear in the journal Designs, Codes and Cryptography. Click here for the parameters of these designs. Yury Ionin (yury.ionin@cmich.edu). Oct. 2003.

Page 79, Known Symmetric Designs (Family 9). There are two series of symmetric designs with parameters (3^m(3^m-1)/2, 3^m-1(3^m +1)/2, 3^m-1(3^m-1 +1)/2) and (3^m(3^m +1)/2, 3^m-1(3^m -1)/2, 3^m-1(3^m-1 -1)/2) respectively. Both series admit the orthogonal groups O(2m+1,3) as groups of automorphisms. In both cases O(2m+1,3) acts on points and lines transitive as a rank 3 permutation group. The representations on points and lines are in both cases equivalent. U. Dempwolff (dempwolff@mathematik.uni-kl.de) Jan. 1999.

Page 80, Section I.5.6. A new symmetric design with parameters (105,40,15) has been found by Zvonimir Janko. A description of it can be found here. (janko@mathi.uni-heidelberg.DE). November 25, 1996.

Page 80, Section I.5.6. There are at least 52432 symmetric (100,45,20) designs on which Frob_10 x Z_2 acts as an automorphism group. These designs correspond to 52432 Bush-type Hadamard matrices. Each Bush-type Hadamard matrix leads to an infinite class of twin designs with parameters v=100(81^m + ... + 81 + 1), k=45(81)^m, lambda=20(81)^m, and Siamese twin designs with parameters v=100(121^m + ... + 121 + 1), k=55(121)^m, lambda=30(121)^m, where m is an arbitrary positive integer. Dean Crnkovic and Dieter Held (deanc@pefri.hr, held@mathematik.uni-mainz.de). February 2001.

Page 80, Section I.5.6. There exists a symmetric design with parameters (144,66,30) whose full automorphism group is isomorphic to Aut M_12, the extension of the sporadic simple Mathieu group on 12 letters by an automorphism of order two. The design can be obtained as follows: Let G=M_12 . In G there are exactly 144 subgroups isomorphic to PSL(2,11); denote them by L_1 , ... , L_144 . Define an incidence structure D=(P,B,I) as follows: P = { P_1 , ... , P_144 } , B = { B_1 , ... , B_144 } , P_i I B_j <=> | L_i \cap L_j | = 10 ; 1 <= i,j <= 144 . Then D is a symmetric (144,66,30) design. The full automorphism group of D is Aut M_12 and has order 190,080. Wolfram Wirth (wirth@mathematik.uni-kl.de).Oct. 1999.

Page 80, Section I.5.6. Two additional symmetric design with parameters (144,66,30) have been found. Click here for the construction. Wolfgang Lempken (lempken@werner.exp-math.uni-essen.de) December 1999.

Page 80, Section I.5.6. A new symmetric design with parameters (189,48,12) has just been found by Zvonimir Janko. A description of it can be found here. (janko@mathi.uni-heidelberg.DE). March 1997.

Page 80, Section I.5.6. There are exactly 54 symmetric (196,91,42) designs admitting an automorphism group isomorphic to Frob_{13*6} x Z_3 acting with orbit size distribution (1,13,13,13,39,39,39,39) for blocks and points. For 50 of these designs the full automorphism group has order 234 and is isomorphic to Frob_{13*6} x Z_3. The remaining four design have Frob_{13*6} x Frob_{7*3} as full automorphism group. Among these designs there are 18 self-dual designs and 18 pairs of mutually dual ones. The derived designs (with respect to the fixed block) of the four designs with full automorphism group of order 1638 are cyclic. Dean Crnkovic (deanc@mapef.pefri.hr). May 2005.

Page 80, Section I.5.6. There are exactly 10 symmetric (324,153,72)-designs admitting an automorphism group isomorphic to Frob_{19*9} x Z2 acting with orbit distribution 1,19,38,38,38,38,38,38,38,38. For eight of these designs the full automorphism group has order 342 and is isomorphic to Frob_{19*9} x Z2, one design has Frob_{19*9} x Z4 and the remaining design Frob_{19*9} x Z8 as full automorphism group. All 10 designs are self-dual. The derived design (with respect to the fixed block) of the design with the full automorphism group of order 1368 is 1-rotational. Dean Crnkovic and Dieter Held (deanc@mapef.pefri.hr, held@mathematik.uni-mainz.de). April 2004.

Page 80, Section I.5.6. There is a symmetric (14080,1444,148) design which admits the sporadic simple Fischer group Fi(22) as a group of automorphisms. Fi(22) acts transitive as a rank 3 permutation group on points and lines. The two permutation representations are inequivalent. U. Dempwolff (dempwolff@mathematik.uni-kl.de) Jan. 1999.

Page 90, Theorem 6.17. For k even, if v = k mod (k(k-1)) and v > exp(exp(k^18k^2), then a (v,k,1) RBIBD exists. (The existence of resolvable BIBD with k even and lambda = 1, Chang Yanxun, preprint).

Page 90, Table 6.19. When k=5 and lambda = 1, replace 4905 by 645. R. Julian R. Abel (julian@maths.unsw.edu.au). May 1997.

Page 90, Table 6.19. (v,5,2) RBIBDs exist for all v=5 mod 10 except for v=15 and possibly for v in {45,115,135, 195,215,225, 235,295,315, 335,345,395}. R. Julian R. Abel (julian@maths.unsw.edu.au). March 1998.

Page 90, Table 6.19. When k=5 and lambda=4, the value 15 should be added to the Exceptions. Reference: P. Kaski and P. R. J. Östergård, "There exists no (15,5,4) RBIBD", J. Combin. Des. 9 (2001), 227-232. Patric Östergård (Patric.Ostergard@hut.fi), May 2001.

Page 91, Table 6.21. 45, 225, 345, 465, 645 are the only possible exceptions for (v,5,1) RBIBDs (Abel, Greig) R. Julian R. Abel (julian@maths.unsw.edu.au). May 1997. and M. Greig (greig@sfu.ca) Nov. 1997.

Page 91, Tables 6.21 and Table 6.22. (75,5,2) and (185,5,1) RBIBDs exist. M. Greig (greig@sfu.ca) Nov. 1997.

Page 91, Table 6.22. A (15,5,4) RBIBD does not exist. Reference: P. Kaski and P. R. J. Östergård, "There exists no (15,5,4) RBIBD", J. Combin. Des. 9 (2001), 227-232. Patric Östergård (Patric.Ostergard@hut.fi), May 2001.

Page 91, Table 6.23: (v,6,5) RBs have been found for v= 48, 60, 138, 318, 330, 336, 414 and 594. R. Julian R. Abel (julian@maths.unsw.edu.au). April 1999 and April 2000.

Page 91, Table 6.25. A (616,8,7) RBIBD exists. M. Greig, Nov. 1997. A (504,8,7) RBIBDs exist. M. Greig (greig@sfu.ca) Aug. 1999.

Page 91, Table 6.26. A (145,6,5) NRB exists. So do (v,7,6) NRBs for v= 85, 99, 141, 148, 589, 596, 603. R. Julian R. Abel (julian@maths.unsw.edu.au). Sept. 1998.

Page 91: Table 6.26: (v,6,5) NRBs exists for all v = 1 mod 6. (v,7,6) NRBs exists for all v=1 mod 7 except possibly for v=183,246,267,274,295. (v,8,7) NRBs exists for all v=1 mod 8 except possibly for v=161,321,385,481,553. R. Julian R. Abel (julian@maths.unsw.edu.au). April, 1999.

Page 91, Table 6.26. A (545,8,7) NRB exists as does a cyclic (129,8,7) NRB. R. Julian R. Abel (julian@maths.unsw.edu.au). March 1998 and Oct. 1998.

Page 91, Table 6.26: (v,8,7) NRBs exist for 161, 321, 481, leaving 385, 553 the only open cases. R. Julian R. Abel (julian@maths.unsw.edu.au). August 1999.

Page 93, Reference [1] has appeared in Des. Codes Cryptogr. 11 (1997) 123--140. (greig@sfu.ca) Aug. 1999.