# New Results in Part IV

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 IV of the Handbook (pages 227-514).

Last edited on 6/10/05.

Page 233, Table 2.2. Kunkle, Sarvate and Greig have pointed out several errors in Table IV.2.2, most of which are inconsistencies for V=K designs. Recent work, including unpublished work by Greig, has resulted in a number of improvements to the table. For an updated version of the table, contact greig@sfu.ca.

Page 243, Theorem 4.14: A BRD(v,4,2) exists for all v\equiv 1 (mod 6), not just prime powers. Click here for a cryptic justification of this statement. M. Greig (greig@sfu.ca) Jan. 1999.

Page 243, Theorem 4.16. The BRD(v,4,2t) possible exceptions can be limited to t=3,5,7 with v=28. A BRD(34,4,6) can be obtained from the {4,1}-GDD of type 2^{34} given by Brouwer et al. [Disc. M. 20(1977), 1-10]. A B({5,6,7},1;39) formed by removing a triangle of lines in PG(2,7), except for one point per line, with these 3 points non-collinear deals with the v=39 case. M. Greig (greig@sfu.ca) Jan. 1999. The exceptions for v=28 can also be removed, using two copies of the Wh(28) on p. 506. In the first copy, use the whist property to assign team-mates the same sign, and opponents opposite signs. In the second copy the mixed sign base blocks are 1, 2, 4 and 5, with sign patterns {+ - + +}, {+ + - -}, {+ + - +}, {+ + - -}. M. Greig (greig@sfu.ca) Mar. 2000.

Page 244, Theorem 4.22. This theorem has been extended. Let F be a finite right nearfield of size q and let t be an integer $≥$ 2. Then there is a BGW((q^t-1)/(q-1),q^{t-1}, q^{t-2}(q-1)) over the group F^{*}. See "A difference matrix construction and a class of balanced generalized weighing matrices", V.C.Mavron, T.P.McDonough and C.A.Pallikaros, Archiv der Mathematik, 76 (2001) 259-264. An immediate consequence: The complement of F[t] is signable over any quotient group of F^{*}, for any finite right nearfield F and any integer t 2, where F[t] denotes the symmetric 2-(q^t-1)/(q-1),(q^{t-1}-1)/(q-1),(q^{t-2}-1)/(q-1)) design in Mavron, A class of symmetric designs, Geometriae Dedicata, 1 (1983) 468-476. T. P. McDonough (tpd@aber.ac.uk) Dec. 2001.

Page 259, Table 7.29. In April 2004, it was announced at the IEEE Radar Conference that 200 distinct Costas arrays exist for n = 24, and 88 exist for n = 25. (See http://www.radar04.org/program/index.cgi?paper=144 ).

Page 260, Section IV.8. Dan Gordon (gordon@ccrwest.org) is maintaining a web page with (most of) the best known covering designs for v,k,t and the number of blocks not too large. The page is at http://www.ccrwest.org/cover.html. It contains a number of improvements on the tables in this section of the Handbook of Combinatorial Designs.

Page 261, Section IV.8.4. It has been shown that the Schonheim bound (Theorem IV.8.6) is true with equality for v sufficiently large in the case when lambda = 1 and t = 2. That is, C(v,k,2) = L(v,k,2) for v sufficiently large, except in the case when is k is odd, (k-1) divides (v-1) and v(v-1)/(k-1) + 1 = 0 (mod k), in which case C(v,k,2) = L(v,k,2)+1. This is a corollary of a more general theorem that determines H-coverings of K_v for any graph H. Yair Caro and Raphy Yuster (ZEAC603@uvm.haifa.ac.il).

Page 262, Theorem 8.13. An optimum (757,5,2) covering can be constructed using Abel and Greig's (165,5,1) RBIBD. The largest open 17 mod 20 case is now 637. Other (v,5,1) RBIBDs and new PBD(v,{5,29*}) (from III.1.38 and Abel and Ling's improvement) also give new (20m+9,5,2) optimum coverings. M. Greig (greig@sfu.ca) Nov. 1997.

Page 262, Theorem 8.14. An optimum (753,5,2) covering can be constructed using Abel and Greig's (165,5,1) RBIBD. The largest open 13 mod 20 case is now 633. M. Greig (greig@sfu.ca) Nov. 1997.

Page 262, Table 8.20: Update the table on the right with the following values for t=3, k=5: Uenal Mutlu (bm373592@muenchen.org).

 v=14 <= 43 15 <= 57 16 <= 67 19 <= 113 20 <= 138

Page 262, Table 8.20: When t=3, k=6, update the table on the right with the the new value for v=20 of <= 74. and when t=3, k=7, update the table on the right with the new value for v=20 of <= 50. Uenal Mutlu (bm373592@muenchen.org).

Page 262, Table 8.21. It should be noted that In the exceptional cases only one extra block is needed. (This is not a new result; it's in Mills' paper [5]). Adding this sentence shows the exceptions are genuine (which might be inferred), and also says what the covering number actually is for the exceptions). M. Greig (greig@sfu.ca) Nov. 1997.

Page 263, Table 8.25: There are many new values reported for this table. We will put a new table below with these new values. Note that for t=4, k=6 all entries in the existing table need to be moved down one cell. (See http:// www.emba.uvm.edu/~dinitz/errata.html ). Thanks to Uenal Mutlu (bm373592@muenchen.org) for these updates.

 t=4 v\k 5 6 7 8 14 <= 232 <= 80 <= 25 15 <= 300 <= 120 <= 57 16 <= 419 <= 160 <= 86 17 <= 492 <= 210 <= 117 18 <= 671 <= 252 <= 152 19 <= 850 <= 343 <= 182 <= 94 20 <= 1096 <= 400 <= 225

 t=5 v\k 6 7 8 9 13 <= 78 14 <= 378 <= 144 <=65 15 <= 610 <= 203 <= 105 16 <= 808 <= 321 <= 117 <=74 17 <= 1227 <= 463 <= 200 <=90 18 <= 1548 <= 663 <= 277 <=145 19 <= 2211 <= 824 <= 408 <= 188 20 <= 2900 <= 1153 <= 526 <=261

Page 266, Table 9.2. Line 1 of this table can be replaced by the following: all, 1, all (m,1)-admissible (i.e. it is known that there exist m-cycle systems with lambda = 1 for all admissible n). The references for this are B. Alspach and H. Gavlas, Cycle decompositions of Kn and Kn - I, J. Combin. Theory Ser. B, 81 (2001) 77-99 and M. Sajna, Cycle decompositions III: Complete graphs and fixed length cycles. J. Combin. Des. 10 (2002) 27-78. Heather Gavlas (hgavlas@emba.uvm.edu) March 2002.

Page 273, Examples 10.10. The following is a (37,6,5) cyclic difference family: {0 1 2 11 13 16}, {0 1 5 10 13 28}, {0 1 8 17 19 24}, {0 3 5 11 20 23}, {0 3 7 13 15 19}, {0 6 7 13 17 21}. Ian Wakeling (ian@qistatistics.co.uk), July 2000.

Page 274, Theorem 10.15 (also mentioned in the errata page). Here is a variant of the construction given in the Handbook. Suppose q = 7 (mod 12) is a prime power and there exists a cube root of unity omega in Fq such that x = omega - 1 is a primitive cube but not a square. Let n be the minimum positive integer such that xn/(x-1) is a 6th root of unity in Fq Then, if n is even, the following base blocks form a (7q,4,1)-DF over Z7 x Fq:
1. {(0,0),(0,x^i),(0,x^i*omega),(0,x^i*omega^2)} for (0 < i < n with i even) and (n < i < (q-1)/6 with i odd).
2. {(0,0),(1,s),(2,s*omega),(4,s*omega^2)} for s a square of F_q distinct from 1.
3. {(0,0),(0,-omega^i),(0,-x*omega^i),(2^i,0)} for i=0,1,2.
M. Buratti(buratti@mat.uniroma1.it), Dec. 2002.

Page 274, Remark 10.16. The first value of q for which the above up date gives a (7q,4,1)-DF not captured by Theorem 10.15 is 283. M. Buratti(buratti@mat.uniroma1.it), Dec. 2002.

Page 274, Remark 10.19. For k = 4, 5, or 6, there exists a (q,k,1) difference family for any admissible prime power q with the only exception of (q,k) = (61,6). This has been proved in the papers: K. Chen and L. Zhu, Existence of (q,k,1) difference families with q a prime power and k = 4, 5, preprint and K. Chen and L. Zhu, Existence of (q,6,1) difference families with q a prime power, preprint. Reported by M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 281, Theorem 10.53. In part 5, a (577,9,1) difference family was found by M. Buratti; (julian@maths.unsw.edu.au), May 1997.

Page 281, Theorem 10.55. There exists a cyclic (4v,4,1)-BIBD for any v of the form v=p1p2...pn where each pi is a prime congruent to 1 (mod 6) such that (pi-1)/6 has a prime factor not greater than 19. M. Buratti (buratti@mat.uniroma1.it), April 2001.

Page 281, Theorem 10.55. This has been extended to all primes p = 1 (mod 12) (M. Buratti, Some abelian Steiner 2-designs of block size 4, preprint). (buratti@mat.uniroma1.it), Sept. 1997.

Page 281, Theorem 10.55 It has been shown that a (4p^{alpha},4,1) elementary resolvable partial difference family over the cyclic group exists. The definition of an elementary resolvable partial difference family can be found in M. Buratti: On resolvable difference families, Des. Codes Cryptogr. 11 (1997), 11 - 23. This improvement is contained in the following paper: C. Lam and Y. Miao, On cyclically resolvable cyclic Steiner 2-designs, J. Combin. Theory, Series A, submitted, 1997. Ying Miao (ymiao@cs.concordia.ca). March 1998.

Page 282, Remark 10.56. If q is a prime power and if k or k - 1 divides 2 x lambda, then there exists a (kq, k, lambda) partial difference family over Z_k x F_q provided that lambda(q - 1) = 0 mod k(k - 1). (See C. Lam and Y. Miao, (C_k + G, k, lambda) difference families, preprint, 1998). Ying Miao (ymiao@cs.concordia.ca). March 1998.

Page 282, Theorem 10.59. This has been improved to the following: If k, q are odd prime powers such that k = 1 (mod 4) and q = 1 (mod k-1) then there exists a (kq,k,(k-1)/4) "partial" difference family over F_k x F_q (M. Buratti, Old and new designs via difference multisets and strong difference families, preprint). In this paper there are several other infinite families of partial difference families and 1-rotational difference families.(buratti@mat.uniroma1.it), Sept. 1997.

Page 282, Theorem 10.60. For k = 7, this has been improved to ANY q (instead of q < 5000) q \neq 19,25,37. For k = 9, this has been improved to ANY q (instead of q < 5834), q \neq 17,25,41,97,113. M. Buratti (buratti@mat.uniroma1.it), July 1999.

Page 282, There exists a (6p,6,1) partial difference family for any prime p = 1 (mod 30). For p not equal to 61, this is a consequence of the improvement of Remark 10.19 combined with the following theorem (M. Buratti, From a (G,k,1) difference family to a (C_k+G,k,1) difference family, DCC 11 (1997), 5-9). "If p is a prime and there exists a (p,k,1) difference family, then there exists a (kp,k,1) partial difference family". Also, a (6*61,6,1) partial difference family was found by M. Greig (M. Greig, Some GDD's constructions, JCMCC 27(1998), 33-52). M. Buratti (buratti@mat.uniroma1.it), Oct. 1997.

Page 282, Remark 10.61. The k=7,9,11 families for Theorem 10.60 are now published in JCMCC 27(1998) 33--52. (greig@sfu.ca) Aug. 1999.

Page 282, Theorem 10.63. This theorem gives the necessary and sufficient condition for the existence of a 1-rotational STS(v) over the CYCLIC group. The necessary and sufficient condition for the existence of a 1-rotational STS(v) over an ABELIAN group is the following: v = 3 or 9 (mod 24) or v = 1 or 19 (mod 72). (Ref: M. Buratti, "1-rotational Steiner triple systems over arbitrary groups", preprint.) In the above paper there are also constructions for 1-rotational Steiner triple systems over NON-ABELIAN groups. M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 282, Theorem 10.64. This has been improved to the following: There exists a (cyclically resolvable) 1-rotational (3v+1,4,1) design whenever all the primes dividing v are equivalent to 1 (mod 4) (without any further hypothesis). (S. Kageyama and Y. Miao, Note on a paper "1-rotational designs with block size 4, Bulletin of the ICA 13 (1997), 82-84). Also, in the case k=5, there exists a 1-rotational (4q+1,5,1) design over Z2 x Z2 x Zp for any prime p = 1 (mod 30) such that (11+5*Sqrt5)/2 is not a cube mod p (M. Buratti, Some constructions for 1-rotational BIBD's with block-size 5, Australasian J. Combin., to appear). Reported by M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 282, Theorem 10.64. Using the new 1-rotational (49,4,1)-BIBD's over (Z_3)+(Z_4)+(Z_4) (see update on Design #146), by means of a recursive construction we get the existence of a 1-rotational (48q+1,4,1)-BIBD over (Z_3)+(Z_4)+(Z_4)+(F_q) for any prime power q = 1 (mod 12). (Ref: M. Buratti and F. Zuanni, "On singular 1-rotational difference families", preprint.) M. Buratti (buratti@mat.uniroma1.it), Dec. 1997.

Page 288, Theorem 11.4. Let G be any group of order g. Then there exists a (g,p;1) difference matrix over G where p is the smallest prime dividing g. (This was previously known only when G is abelian). Also Let p be a prime and let G be an abelian group of order p^e and type (n_1,n_2,...,n_t). Then there exists a (p^e,p^f;1) difference matrix over G where f = Floor[(n_1+ n_2 +...+ n_t)/max(n_1,n_2,...,n_t)]. Both results appear in (M. Buratti, Recursive constructions for difference matrices and relative difference families, JCD, to appear). M. Buratti (buratti@mat.uniroma1.it), Oct. 1997.

Page 289, Theorem 11.15. The following is a supplementary result which limits the type of group if the matrix has a structural group-theoretic property. Let H be a generalized Hadamard matrix over a group G. Suppose H has a row w, whose entries are not all equal, with the property that if r is any row of H, then rw is also a row of H. Then G is elementary abelian. See "Generalised Hadamard Matrices and Translations", T. P. McDonough, V. C. Mavron and C. A. Pallikaros, Journal of Statistical Planning and Inference 86 (2000) 527-533. T. P. McDonough (tpd@aber.ac.uk) Dec. 2001.

Page 291, Theorem 11.44: This theorem is generalized to: Let H be a multiplicative group of order n. Suppose that there is a v x w matrix M=[m_{i,j}], with entries m_{i,j}\in\widehat H, satisfying the following conditions: (i) the number of entries of each row which are different from 0 is a constant k; (ii) there is a constant $λ$ such that, in the difference list of two distinct rows, the element 1 appears $λ$-1 times and each h \in H-{1} appears $λ$times. Further, suppose that (n+1)(2(k+1)-(n+1)$λ ≥ nw$ . Then, for any group G of order n+1, there is a difference matrix over G with v rows and 2(n+1)(k+1)-(n+1)^2$λ$ columns and of index 2(k+1)-(n+1)$λ$. See "A difference matrix construction and a class of balanced generalized weighing matrices", V.C.Mavron, T.P.McDonough and C.A.Pallikaros, Archiv der Mathematik, 76 (2001) 259-264. T. P. McDonough (tpd@aber.ac.uk) Dec. 2001.

Page 314, Section 14.3: A web page ( http://www.research.ibm.com/people/s/shearer/dtsub.html) is being maintained which lists the best difference triangle sets currently known. James B. Shearer (JBS@watson.ibm.com). Jan. 1999.

Page 315, Table 14.15: The paper "Application of Linear Programming to the Optimal Difference Triangle Set Problem" (by R. Lorentzen and R. Nelson, IEEE Transactions on Information Theory, 37(1988), p. 1486- 1488) has better lower bound values for many cases with n=1,...,5. James B. Shearer (JBS@watson.ibm.com). Jan. 1999.

Page 316, Table 14.18: Two new Golomb rulers have been found: m(1,19)=283 and m(1,20)=333 are now known to be exact. See http://members.aol.com/golomb20/index.html) for this and other information about the search for Golumb rulers. James B. Shearer (JBS@watson.ibm.com). Jan. 1999.

Page 324, Table 17.5: There is an improvement in row 12 of this table. Darryn Bryant has proven that any partial Steiner triple system of order u can be embedded in a Steiner triple system of order v if v 3u-2 and v 1 or 3 (mod 6). Reference: Darryn Bryant, Embeddings of partial Steiner triple systems. J. Combin. Theory Ser. A 106 (2004), no. 1, 77--108.

Page 355, Section 20.2. A table of lower bounds for the number of mutually orthogonal frequency squares has been published. The reference is: C. Laywine and G. Mullen, A table of lower bounds for the number of mutually orthogonal frequency squares, Ars Combinatoria 59 (2001), pp. 85 - 96. July 2001.

Page 355, Theorems 20.8-11: In "The Geometry of Frequency Squares", D. Jungnickel, V. C. Mavron and T. P. McDonough, Journal of Combinatorial Theory (A) 96 (2001) 376-387, a correspondence between mutually orthogonal frequency squares (MOFS) and nets satisfying an extra property ("framed nets") is established. Theorems 20.10 and 20.11 are contained in the result: There exists a set of r MOFS of type F(n;\mu) if, and only if, there exists a framed (m,r;\mu^{2})-net, where n=\mu m. The concepts involved are described as follows: Let D be any design with n^{2} points and \mu n points on each block. A frame [X:Y] of D consists of two partitions X={X_{1},X_{2},...,X_{n}} and Y={Y_{1},Y_{2},...,Y_{n}} of the points of D into subsets of order n, such that for all i,j: (a) |X_{i}\cap Y_{j}|=1; (b) |X_{i}\cap B|=|Y_{j}\cap B|=\mu for all blocks B. D is a framed design if it has a frame. The net corresponding to a complete set of MOFS is a complete-framed net. The constructions in Theorems 20.8 and 20.9 can all be obtained by the construction implicit in the this theorem. These are contained in the following results:
(A) An (m,r;1)-net is framed if, and only if, it can be extended by 2 parallel classes; that is, it is a subnet of some (m,s;1)-net, where s\geq r+2.
(B) Any framed (m,m-1;1)-net is embeddable in an affine plane of order m. There is a unique non-framed (m,m-1;1)-net and this net has m=4.
(C) Suppose there exists a framed (m,r;\mu^{2})-net or, equivalently, a set of r MOFS of type F(n;\mu), where n=\mu m and m>1. Then r\leq (n-1)^{2}/(m-1) with equality if, and only if, the net is a PBIBD based on the L_{2}-association scheme. In this case, \lambda_{1}=(n-1)(\mu-1)/(m-1) and \lambda_{2}=(\mu n-2\mu+1)/(m-1).
(D) Let D be a PBIBD based on the L_{2}-association scheme and suppose D is an (n/\mu,r;\mu^{2})-net. Then D is a complete-framed net if, and only if, \lambda_{1}=r(\mu-1)/(n-1) and \lambda_{2}=r(\mu n-2\mu+1)/(n-1)^{2}. In this case, r=(n-1)^{2}/(m-1).
(E) If there exists a framed (m,r;\mu^{2})-net or, equivalently, a set of r MOFS of type F(\mu m;\mu), and there exists an (m,s;\mu/m)-net, then there exists an (m,r+2s;\mu^{2})-net.
(F) Suppose there exists an (m,r;\mu/m)-net \Gamma and an (m,s;\mu)-net \Delta and the dual of \Delta is resolvable. Then: (a) There exists a framed (m,r(s-1);\mu^{2})-net and therefore a set of r(s-1) MOFS of type F(\mu m;\mu). (b) The set of MOFS in (a) is complete if and only if \Gamma is a complete net and \Delta a symmetric net.
(G) Suppose there exists an (m,r;\mu)-net \Gamma and a (\mu m^{2},s;\nu^{2})-net \Delta. Then there exists an (m,rs;\mu^{2}m^{2}\nu^{2})-net \Sigma. If \Delta is framed then so is \Sigma. In this case \Sigma is complete-framed if, and only if, \Delta is complete-framed and \Gamma is a complete net.
(H) If there exists an (m,r;\mu)-net and a set of s MOFS of type F(\mu\nu m^{2};\nu), then there exists a set of rs MOFS of type F(\mu\nu m^{2};\mu m\nu). The second set of MOFS is complete if, and only if, the first set is complete and the net is complete. T. P. McDonough (tpd@aber.ac.uk) Dec. 2001.

Page 363, Theorem 22.15. This can be updated to: The necessary conditions for the existence of a (Kn, Ck)-design are sufficient for all k and n. The references are B. Alspach and H. Gavlas, Cycle decompositions of Kn and Kn - I, J. Combin. Theory Ser. B, 81 (2001) 77-99 and M. Sajna, Cycle decompositions III: Complete graphs and fixed length cycles. J. Combin. Des. 10 (2002) 27-78. Heather Gavlas (hgavlas@emba.uvm.edu) March 2002.

Page 363, Theorem 22.17. (a) There exists a cyclic (Kv,Ck)-design for any v = 1 (mod 2k). (b) There exists a cyclic (Kv,Ck)-design for any v = k (mod 2k) and k odd with the only definite exceptions of (v,k)=(9,3); v=k=15; v=k=pa with p prime and a > 1. Marco Buratti (buratti@mat.uniroma1.it), March 2003.

Page 363, Theorem 22.17. There exists a 1-rotational (K_{2kn+1},C_k) design if and only if k is an odd composite number. There exists a 1-rotational (K_{2kn+k},C_k) design if and only if k is odd but n \not\equiv 2,3 (mod 4) when k=3. Marco Buratti (buratti@mat.uniroma1.it), March 2002.

Page 363, Theorem 22.17. There exists a 1-rotational (Kv,Ck) design for all admissible pairs (v,k) with k odd and k v < 3k but v 2k+1 when k is a prime. The construction is very easy. Hence, in view of Theorem 22.16, it gives a very quick proof of the theorem by Alspach and Gavlas (see update of Theorem 22.15) in the case of k odd. Marco Buratti (buratti@mat.uniroma1.it), August 2002.

Page 363, Theorem 22.22. There exists a (K_n,Q_k)-design for any n of the form n=k2^{k+e}t + 2^e for any triple of non-negative integers (k,e,t) with k e and 2e = 1 (mod k). Marco Buratti (buratti@mat.uniroma1.it), Nov. 1999.

Page 363, Theorem 22.25(1): This theorem can be improved as follows: Let n 5 be an integer and let n= (2^e)(p_1^e_1)(p_2^e_2) ...(p_k^e_k) (with e 0 and e_i 1) be its decomposition into prime factors. Furthermore, suppose that for all odd e_i it holds that p_i = 1(mod 4) and p_i < 100000. If e > 2 or e=0, then all 2K_{zn} have a self-orthogonal factorization into hamiltonian paths when z =1 or 3 or 5 <= z <= 101. If e = 1 and e_i 2 for some i, then 2K_n has a self-orthogonal factorization into hamiltonian paths. V. Leck Ph.D thesis, University of Rostock, 2000. (volker.leck@mail1.uni-rostock.de). June 2000

Pages 364/365, Table 22.34. Some exceptions may be removed: A (K_{65},K_5 - P_3)-design exists (K_5 - P_3 is the 21st graph of the table). For p=37, 73, 109, 397, 487, 541, 613 a (K_p,K_5 - e)-design exists (K_5 -e is the 23rd graph of the table). Marco Buratti (buratti@mat.uniroma1.it), Nov. 1999.

Page 371, Remark 24.15. Let p and 2p-1 be prime powers and p=3 (mod 4). Then there exists a symmetric design with parameters (4p^2, 2p^2 - p, p^2 - p) (from Page 79, Family 6). Thus there exists a regular Hadamard matrix of order 4p^2. In particular, there exists a regular Hadamard matrix of order 4*79^2=24964. Dean Crnkovic (deanc@pefri.hr). April 2005.

Page 373, Table 24.31: Dokovic has Goethals-Seiden type constructions for n = 81, 103, 151, 169, 463 [Australasian JC 10 (1994) 259-264]. M. Greig (greig@sfu.ca). Jan. 1998.

Page 374, Table 24.32. A symmetric Hadamard of order 4*81 exists. Use Ehlich's construction of Theorem 14.4.1 in [Hall, 2nd ed.] and replace G by GP and I_{19} by P, where P is the permutation matrix that fixes the first column and reverses the rest, and take G as the Paley quadratic indicator matrix. M. Greig (greig@sfu.ca) Nov. 1997.

Page 374, Table 24.33. A Hadamard matrix exists for orders 4*8281, 4*9409, 4*9801. Use Ehlich's construction of Theorem 14.4.1 in [Hall, 2nd ed.]. M. Greig (greig@sfu.ca) Nov. 1997.

Page 387, Table 27.13. The following are updates for this table: 2 Max MOLS(n) for n=16,28,31,34,46; 3 Max MOLS(n) for n=13,17,33,40,49,57; 4 Max MOLS(n) for n=36,46; 5 Max MOLS(49); 6 Max MOLS(50); 7 Max MOLS(57). These values are from the paper by David Drake, John van Rees and Wal Wallis, Maximal sets of mutually orthogonal latin squares, [Discrete Math 194 (1999), 87-94]

Page 387, Table 27.13. 3 Max MOLS(15), 3 Max MOLS(16), 4 MaxMOLS(12), 4 MaxMOLS(16), 4 MaxMOLS(24), 4 MaxMOLS(28) all exist. These values are from the paper New and old values for maximal MOLS(n), by David Bedford and Roger M. Whitaker. Roger M. Whitaker (mad09@cc.keele.ac.uk). June 1998.

Page 391, Table 28.17. (v,5,1) PMDs exist for all v=0,1 mod 5 except 6, 10 and possibly v=15, 20. For lambda > 1 and k=5, the necessary conditions v 5 and lambda v(v-1) = 0 mod 5 are sufficient. (Bennett, Y.Chang, Yin, H.Zhang). R. Julian R. Abel (julian@maths.unsw.edu.au). May 1997.

Page 391, Table 28.17. The largest unknown (v,6,1) PMDs for v = 0,3,4 mod 6 are 300, 657, 148 respectively. There are no possible exceptions for v = 1 mod 6. (Abel, Bennett, H. Zhang) R. Julian R. Abel (julian@maths.unsw.edu.au). August 1998. Frank Bennet reports that the largest possible exception for the case v = 0 (mod 6) is now v = 198. Frank Bennett (FBENNETT@LINDEN.MSVU.Ca). May 1999.

Page 391, Table 28.17 (and Table 28.18) The necessary conditions for existence of a (v,6,lambda)-PMD are v 6 and either lambda = 0 (mod 3) or v = 0 or 1 (mod 3). For lambda > 1, these are sufficient, except possibly for v=6 and either lambda = 2 or lambda odd. R. Julian R. Abel (julian@maths.unsw.edu.au). April 2005.

Page 391, Table 28.19. (v,7,1)-PMDS exist for v=105, 266, 267, 274, 315, 322, 364. In addition, if lambda > 1 and v = 0 or 1 mod 7 then a (v,7,lambda)-PMD exists if v is not 42 (Bennett, Abel). R. Julian R. Abel (julian@maths.unsw.edu.au). May 1997.

Page 391, Table 28.19. (v,7,1)-PMDS exist for v = 168 = 7 x 24, v = 252 = 7 x 36 and v = 253 = 7 x 36 + 1. These come from the existence of 5 idempotent MOLS(n) for n = 24 and 36. Frank Bennett (FBENNETT@LINDEN.MSVU.Ca). May 1999.

Page 391, Table 28.19. (v,7,1)-PMDS exist for v= 378 and 420. (Abel, Bennett) R. Julian R. Abel (julian@maths.unsw.edu.au). Sept. 1997. Also, (v,7,1) PMDs exist for v=85, 106 and 155. R. Julian R. Abel, Sept. 1999.

Page 392, Table 28.22. There are exactly 4,905,693 inequivalent MTS(12)'s. So the table should now be as follows:

 v 3 4 6 7 9 10 12 M(v) 1 1 0 3 18 143 4,905,693
(Note the correction to v=10). More informaion on the enumeration of the MTS(12)'s can be found here. Paul Denny (pden001@cs.auckland.ac.nz). Jan. 1998.

Page 410, Section IV.33.3. It has been shown that the First Johnson bound (Theorem IV.33.5) is true with equality for v sufficiently large in the case when lambda = 1 and t = 2. That is, D(v,k,2) = U(v,k,2) for v sufficiently large, except in the case when (k-1) divides (v-1) and v(v-1)/(k-1) (mod k) is not 0, in which case D(v,k,2) = U(v,k,2)-1. This is a corollary of a more general theorem that determines H-packings of K_v for any graph H. This has been published in the Electronic Journal of Combinatorics; the URL of the abstract is http://www.combinatorics.org/Journal/Volume_4/Abstracts/v4i1r1.html. Yair Caro and Raphy Yuster (ZEAC603@uvm.haifa.ac.il).

Page 419. Kishore Sinha's web site contains updated group divisible designs and partially balanced design tables from Clatworthy(1973). In addition there is a nice bibliography.

Page 446, Theorem 40.23. The possible exceptions for k have been improved leaving only k = 6, 8, 10 still undecided. Zhu Lie (lzhu@suda.edu.cn), March 1999.

Page 449, Theorem 41.5. Regular SOLSSOMs exist for n=66, 70. R. Julian R. Abel (julian@maths.unsw.edu.au). April 1999.

Page 451, Theorem 41.16. The necessary conditions for existence of an HSOLSSOM of type h^n, namely, (1) n 5, and (2) n must be odd whenever h is odd, are sufficient, except possibly for h=6 and n = 12 or 18. R. Julian R. Abel (julian@maths.unsw.edu.au). Nov 2000.

P451, Theorem 41.22. 2-SOLSSOMs have been found by J. Abel and F. Bennett for all n > 700. R. Julian R. Abel (julian@maths.unsw.edu.au). Sept. 2002.

Page 452, Remark 41.23. There exists a 2-SOLSSOM(n) for any odd integer n 7 except possibly for n in {15, 21, 33, 35, 39, 45, 51, 65, 87, 123, 135}. R. Julian R. Abel (julian@maths.unsw.edu.au). Sept 2000.

Page 454, Proposition 42.11. Base sequences exist for m=31. (D.Z.Dokovic, Base sequences, complementary ternary sequences, and orthogonal designs, JCD, Vol.4, No.5 (1996), 339-351). C.Koukouvinos (ckoukouv@atlas.uoa.gr). Nov. 1998.

Page 467, before Example 45.3. In the sentence "In any abelian group G of odd order, ... " the word abelian may be deleted. It is easy to prove that if G is an additive group of odd order, then the set P is a starter also in the nonabelian case. See also the paper "One-Factorizations of Complete Graphs with Vertex-Regular Automorphism Groups" by A.Bonisoloi and D.Labbate, J. Comb. Designs 10, (2002), 1-16. Gloria Rinaldi (rinaldi.gloria@unimore.it). May 2004.

Page 468, after Remark 45.12. Generalizing the definition of starter to groups of even order, 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 469, Theorem 45.20. The following is the strongest result on the existence of skew starters: There exists a skew starter in Zv for all v such that gcd(v,6)=1 and v either is not divisible by 5 or is divisible by 25. (see Chen, Ge, Zhu, Starters and related codes. Special issue in honor of Professor Ralph Stanton. J. Statist. Plann. Inference 86 (2000), no. 2, 379--395.) May 2004.

Page 481, Theorem 48.7. Row-complete latin squares of every composite order exist. Ref: J. Higham, "Row-complete latin squares of every composite order exist", J. Combin. Des. 6 (1998), 63-77. J. Higham (jhi456@yahoo.ca). Dec. 1997.

Page 481, Theorem 48.9. If there exists a row-complete latin square (roman square) of order m and a sequenceable group of order n, then there exists a row-complete latin square of order mn. Ref: J. Higham, "A product theorem for row-complete latin squares", J. Combin. Des. 5 (1997), 311-318. J. Higham (jhi456@yahoo.ca). Dec. 1997.

Page 499, Table 52.28. A weighing matrix of order 17 and weight 9 has been constructed by Hiroyuki Ohmori (ohmori@edmath.ed.ehime-u.ac.jp). Also, a weighing matrix of order 87 and weight 49 has been constructed by Yoseph Strassler (strasler@bimacs.cs.biu.ac.il).

Page 499, Table 52.28. A weighing matrix of order 33 and weight 25 has been constructed by K.T. Arasu and D. Torban (KARASU@desire.wright.edu). Sept. 1997.

Page 501, Table 52.29. A weighing matrix of order 34 and weight 18 can be constructed from the new weighing matrix of order 17 and weight 9 that is mentioned above. Yoseph Strassler (strasler@bimacs.cs.biu.ac.il) and Masaaki Harada. (harada@kszaoh3.kj.yamagata-u.ac.jp). March 1999.

Page 505, Theorem 53.8.2. This result can be improved to 4n=2^\alpha (for alpha 2). These designs are actually Z-cyclic TWh(4n), and are constructed by presenting the log table of GF(4n) as an n by 4 table, (with log(0) = \infty), and taking the rows as the games of the first round, with the column pairs defining the team/opponent kinds. M. Greig (greig@sfu.ca) Aug. 1999.

Page 506, Theorem 53.13. There exists a Z-cyclic DWh(2^{2n}+2^n+1) for any n 2. A Z-cyclic DWh(v) also exists for the following values of v: 121, 141, 161, 201, 261, 301, 361, 481, 501, 561, 681, 781, 801, 861, 961, 981, 1001. M. Buratti (buratti@mat.uniroma1.it), July 1999.

Page 507, Theorem 53.16. A TWh(4n+1) exists for all n \geq 1 except possibly n=3, 4, 11, 14, 16, 17, 19, 21, 23, 29, 32, 33, 38. A TWh(4n) exists for all n \geq 1 except possibly n=3, 14. (See (JCD 5(1997) 249-256). Zhu Lie (lzhu@suda.edu.cn), March 1999.

Page 507, Theorems 53.17 and 53.18. There exists a Z-cyclic TWh(p) for any prime p=1 (mod 4) with the only exceptions of p= 5, 13 and 17. M. Buratti (buratti@mat.uniroma1.it), July 1999.

Page 507, Theorem 53.19. There exists a Z-cyclic TWh(3v+1) whenever all the primes dividing v are equivalent to 1 (mod 4). This is a consequence of the improvement to Theorem 10.64 given above. M. Buratti (buratti@mat.uniroma1.it), Sept. 1997.

Page 513, Table 55.10. Nick Phillips (nick@cs.siu.edu) is maintaining a web page where he is keeping the latest information about double Youden rectangles. Click here for that page.