# Handbook of Combinatorial Designs, First Edition

## Errata List

We will be working on this errata list on an ongoing basis. Please send us e-mail concerning any errors that you find in the book so that we can add them to this list.

Thanks to everyone who has pointed out errors, including Julian Abel, Ian Anderson, Juergen Bierbrauer, Marco Buratti, Yeow Meng Chee, Kris Coolsaet, Rob Craigen, Dameng Deng, Paul Denny, Jeff Dinitz, Chin-Mei Fu, Ge Gennian, Don Giles, Malcolm Greig, Jan de Heer, Yury Ionin, Shiro Iwasaki, Dieter Jungnickel, Clement Lam, Reinhard Laue, Volker Leck, Li Maohua, Jooyoung Lee, Vazz Linek, Alan Ling, Mike Maguire, Bill Martin, Brendan McKay, Lucia Moura, Wendy Myrvald, Patric Ostergard, Marco Pavone, David Pike, Alex Pott, Joern Quistorff, Kalle Ranto, Andrew Rimell, Gordon Royle, James Shearer, Doug Stinson, Kirsten Stor, Tran Van Trung, John van Rees, Ian Wakeling, Ian Wanless, Adam Wolfe, Tom Zaslavsky and Hongtao Zhao (so far).

Page vi: Brett Tagendal should be Brett Tangedal.

Page 2, Definition after 1.4: The first aij = 0 should be aij = 1.

Page 7, Table 1.18, Design 2: The 8th block  should be 1369 (not 1349).

Page 12, Table 1.26, Design 6: The 12 block should be 13679cegi (not 13569cegi).

Page 15, Design #44: Nd >=2.2*106 (instead of Nd=22859). Pietsch's error was pointed out by Denny -- see p. 226 of: P.C. Denny, Search and enumeration techniques for incidence structures, CDMTCS Research Report 085, University of Auckland, 1998.

Page 17, Design #153: A (21,6,4) BIBD can be found in Example I.2.41 (on page 46).

Page 19, Design #278: Nd = 8,360,901 (Not 5,201,971).

Page 21, Design 399: Malcolm Greig reports that this design can not be found in the reference given. He states "it is in: R.N.Mohan, A note on the construction of certain BIB designs, Disc.M. 29(1980) 209-211 but it looks like this is not the first instance, as he highlights the fact that the second member of the series kills an open case in a list by Kageyama (in a paper I haven't seen), and #399 is merely mentioned as the first in his series."

Page 23, Design 529: The value of v is incorrect. It should be v=15 (k=6, lambda=10).

Page 40, reference [118]: S. Sillito should be G.P. Sillitto.

Page 46, See Also: The reference for resolvable designs should be I.6, not IV.6.

Page 49, Theorem 3.12: The reference for this result is [67] (the same paper as is referenced in Theorem 3.13).

Page 50, Theorem 3.26: The dual code should have minimum distance f (not e as written).

Page 56, line 17: This was printed badly. It should be 3-(25,4,m2).

Page 56, line 18: This was also printed badly. It should be 3-(25,5,m3).

Page 56, Table 3.37: There exists a 3-(26, 5, 10) design. This is a consequence of Theorem 3.24(2) when q=25 and k=4.

Page 56, Table 3.37: There exists a 3-(26, 8, 4.7) design. This is a consequence of Theorem 3.24(2) when q=25 and k=7.

Page 56, Table 3.37: There exists a 3-(26, 10, 15.3) design. This is a consequence of Theorem 3.24(2) when q=25 and k=9.

Page 57, Table 3.37: There exists a 3-(30,6,5) design. This is a consequence of Theorem 3.24(4) with q = 29.

Page 63, Table 3.44: The first nonexistence bullet should read: If a simple $t-(v,k,\lambda)$ design does not exist (see table 3.37),then a $LS_{\lambda}(t,k,v)$ does not exist.

Page 67, Reference 66: The page number for this paper should be 301-311.

Page 62, line 3: Wasserman should be Wassermann.

Page 69, Construction 4.13: In the last sentence, "$\lambda_{i,j} = \lambda_{i+1,j-1} - \lambda_{i,j-1}." should be "$\lambda_{i,j} = \lambda_{i,j-1} - \lambda_{i+1,j-1}."

Page 76, Remark 5.10: For lambda=3, a symmetric design is also known for k=7 (v=15).

Page 81, (71,15,3) design: The mapping sigma that is given is incorrect. It should be sigma =(A)(K0)(K1 K2 K4)(K3 K6 K5) (3i 52i 74i)(4i 62i 84i), K=1,2,9,10.

Page 85, See Also: The definition of semibiplanes is incorrect as given. It should be ... in which every two points lie on 0 or two common lines, and every two lines meet in 0 or two points.

Page 89, Table of solutions to the Kirkman Schoolgirl Problem. Case 7a-Sunday: the third row should read 'ojfml' instead of 'ofjml'. Also, Case 7b-Sunday: the second row should read 'nidjh' instead of 'nidjn'.

Page 89, Table of solutions to the Kirkman Schoolgirl Problem.: The solutions to the Kirkman Schoolgirl Problem that are numbered 15a and 15b should be 19a and 19b, respectively.

Page 98, Table 1.6: The entry for the number of isotopy classes of latin squares of side 7 should be 564 (not 563 as reported in Denes and Keedwell).

Page 105, Theorem 1.24. The theorem should say if and only of 1 <= n <= 7 or n = 9, 11 or 13. (The 11 was omitted from the original).

Page 108, Theorem 1.48: In the definition of crisscross latin squares there should be a requirement that the square is of even order.

Page 108, Table 1.52: The header in the table should say 2n = (instead of n = ).

Page 112, Example 2.7: The third block in the last column should be {31,42,33,14,25}.

Page 116, Theorem 2.36: In the second square, the digit shown in row 10, column 11 is a 7 - it should be a 2.

Page 116, Theorem 2.37: The "cyclic" latin squares here are different than the "cyclic" latin squares defined on page 444. Here we mean that row i is obtained from row i-1 by adding 1 (mod 15) to each entry of the row.

Page 116, Theorem 2.37: L1 should end 9 8 , not 8 9.

Page 117, Theorem 2.41: Same comment as above except here it is modulo 21.

Page 117, Theorem 2.42: The construction yields a (21,5;1,1;1) - QDM, not a (21,1;1,1;5) - QDM as written.

Page 120, Theorem 2.55: On line -3, Replace t(C) = [(x1,y1,5z5) by t(C) = [(x5,y5,5z5).

Page 121: We inadvertently left out the construction for N(54). There is actually a new result here since the book went to press. Julian Abel has shown that N(54) >= 5. Click here for the construction.

Page 140, Construction 2.83: The statement of the construction should be: If there exists a pair of orthogonal diagonal LS(n), then there exists an nth order magic square.

Page 145, Table 3.10: 4 IMOLS(v,10) are unknown for v=68, 69, 76, and 79. (3 are known for each of these orders). Table 3.12 should be updated accordingly.

Page 174, Table 4.19: The value for n = 6, lambda = 1 should be =3.

Page 174, Table 4.19: The upper bound for n = 10, lambda = 1 should be 9, not 11.

Page 191, Theorem 1.31: 124 should be one of the possible exceptions and 288 should be 268 (See Theorem 50.12, page 491 for the correct statement).

Page 192, Table 1.38: For B(4,5,11*), the exception 139 should be 130 (Bennett).

Page 194, Theorem 2.3 (part 2): The phrase "if there exists PBD(gj+f,K)" can and should be deleted.

Page 204, Example 3.9: For U(K,u), u is an integer such that there exists a K-GDD of type u^k for all k in K (not one of type u^t).

Page 210, Table 3.18: 11,19,23,27,51 should not have been in the exception set for $Q_{\geq4}$, (NB 51=4*11+7 a truncated TD).

Page 211, Table 3.19: In the definition of the three types of sets, replace the > signs by >=.

Page 211, Table 3.19: 31 is not essential in the generating set for K6, because of PG(2,5).

Page 212, Table 3.19:The essential element 15 has been omitted from H{0,1(5)}.

Page 216, Table 4.19: PLIN(15)=119. (There are 120 if the trivial one line space is included.

Page 223, Theorem III.5.13(2). The hypothesis that a (gj+h,K,\lambda)-PBD exists was omitted.

Page 244, Theorem 4.22 (part 3) should read "... and G = Zg-1 where g = 2m-1..." rather than "... and G = Zg where g = 2m-1-1...". (See reference [14]).

Page 258, Example 7.23: Kings should be kings.

Page 260, First Definition: v >= m >= k should be v >= k >= lambda.

Page 261, Remark 8.12: The last line should read: for which it is known that C(v,4,3) = L1(v,4,3) is v = 499.

Page 263, Table 8.25: When t=4 and k=6, the entire column should be shifted down one cell. Specifically, C(6,6,4) = 1, C(7,6,4) = 5, C(8,6,4) = 7, C(9,6,4) = 12, C(10,6,4) = 20, C(11,6,4) = 32, C(12,6,4) <= 41, etc.

Page 263, Table 8.26: "D(v,3,2)" should read "C(v,3,2)" in the table heading.

Page 264, Table 8.27: "D(v,4,2)" should read "C(v,4,2)" in the table heading.

Page 264, Example 8.28: The first block should be {(0,0),(1,0),(1,1),(0,2)}, (not {(0,0),(1,0),(1,1),(2,2)}).

Page 265, Reference [8]: The second author is P.R.J. Ostergard.

Page 272, Examples 10.6: The starter blocks for the 37 should read 0 1 3 24,  0 4 9 15,  0 7 17 25.
(So the 0 4 6 32 block is wrong and should be 0 4 9 15).

Page 274, Theorem 10.15(3): The base block elements (2^t, \omega^{2t}) and (2^t, x.\omega^{2t}) should be (2^t, \omega^t) and (2^t, x.\omega^t).

Page 274, Theorem 10.18: (12t+1,4,1) difference families exist for 3<=t<=23. See Reference [14], p287.

Page 277, Table 10.34: For q=625, the primitive polynomial should be x^4+x^3+x^2+x+3.

Page 280, Table 10.48: The base block {0,1,x,x356} is missing for q=625.

Page 281, Table 10.51: No base block is given for q=197; {0 1 3 9 25 38 80 109} is suitable. (Julian Abel, Sept. 1998)

Page 283, Table 10.69: For v=36, the short block (0 7 14 21 28) is missing.

Page 284, Table 10.70: The second short block for the v=40 design should be (0 2 13 15 26 28)

Page 285, Table 10.72: For (34,11,10) there were 2 incorrect values in the 3rd base block. It should be {0, 1, 2, 6, 9, 10, 13, 15, 23, 27, 32}.

Page 285, Table 10.72: Replace the first and third base blocks by {\infty 0 2 3 5 9 13 21 24 27 32 33} and {2 3 4 6 14 16 22 27 31 35 36 38}. The other 2 blocks (1 2 5 6 7 8 9 15 19 24 29 31) and (0 1 3 9 13 14 16 22 26 27 29 35) should remain unaltered.

Page 285, Table 10.75: The element (4x,x) should be added to the first set of three base blocks.

Page 310, Example 13.20: D14 should read 1 + a + b + ab + c + a2 c.

Page 313, Construction 14.5: After Fq2 add: as an extension over Fq. Also, in line 4 the (cyclic) group has order q2-1, not q2+1.

Page 315, Table 14.15: James B. Shearer (JBS@watson.ibm.com) noted the following errors in Table IV.14.15. "My research report "Some New Difference Triangle Sets" (IBM RC 16610 3/5/91, a version of which has since appeared in The Journal of Combinatorial Mathematics and Combinatorial Computing, 27(1998), p. 65-76) contained numerous examples of difference triangle sets smaller than the bounds in table 14.15. In particular the entries m(3,6)=77, m(4,5)=65 and m(5,5)=83 are wrong. I showed m(3,6)=72, m(4,5)=64 and m(5,5)<=81. These errors were due to a misunderstanding about whether certain computer searches were partial or exhaustive. In other words these entries should have been upper bounds not exact values."

Page 364, Theorem 22.28: isomorphic factors should be isomorphic factors with maximum degree 2

Page 371, Remark 24.15: n <= 428 should be n < 428.

Page 374, Table 24.33: On the second line of the table header, n < 3999 should read n < 9999.

Page 381: The definition of Howell design should begin: Let S be a set of 2n symbols...

Page 391, Table 28.18 The reference for this result should be J. Yin, The existence of (v,6,3)-PMDs, Mathematic Applicata 6, (1993), 457-462 and not [11].

Page 391, Table 28.19: There is no known (28,7,1) PMD and v=153 (after the number 252) should be 253.

Page 392, Table 28.22: Paul Denny (pden001@cs.auckland.ac.nz) reports that for v=10 there are only 143 inequivalent MTS(10)'s, not 144.

Page 394, Example 29.2: 4673 should be 3673.

Page 410, Theorem 33.7: "Suppose d=D_{\lambda}(v,k,t)=qv+r, where ..." should read "Suppose d=D_{\lambda}(v,k,t) and kd=qv+r, where ..."

Page 414 Thmeorem 34.3: (s+1)(st+1)/a should be (s+1)(st/a +1) and (t+1)(st+1)/a should be (t+1)(st/a +1).

Page 420, Remark IV.35.5(2) This remark should have read: The duals of the residual designs of symmetric BIBDs with \lambda=2 are PBIBD(2)s.

Page 438, Table 39.8Room square R5 is equivalent to R6. The correct R5 will appear in the Second Edition of the Handbook. (It can be found in the book by Wallis, Street, Wallis).

Page 443, Theorem 40.7: {0,1} should be {0,1,1/2}.

Page 458, Theorem 43.7: When n=4s, the first pair should be (4s+r-1,8s-r+1). Also when n=4s+1 the pair (4s+r-1,8s-r+3) should be (4s+r+1,8s-r+3).

Page 458, Theorem 43.8 (2.): For a Langford sequence, n = 2,3 (mod 4) and d is even should be n = 0,3 (mod 4) and d is even.

Page 459, Construction 43.11: On the second line, m triples should be n triples.

Page 459, Construction 43.13: When n=4s, the second pair should read (s+r-1,3s-r). (i.e. add a right parenthesis).

Page 482, Remark 48.10: The remark should be "... whenever p < q are odd primes with p | q - 1 and 2 is a primitive root mod p ...". (Reference: A. D. Keedwell, "On the sequenceability of non-abelian groups of order pq", Discrete Math. 37 (1981), 203-216.)

Page 490, reference [13]: PerpendiculAr should be Perpendicular.

Page 491, Theorem 50.5: The condition that r is odd is also needed.

Page 496, Definition of conference matrix: it is necessary to also add that the 0's must be on the main diagonal.

Page 573, Theorem 7.52(1): Should be: then 2n is a multiple of 4.

Page 573, Theorem 7.52(3): Should be: for all odd prime powers q other than Fermat primes. (See Du and Hwang, PAMS 104 (1988), 660-667).

Page 646, Example 3.6: The smallest graph with trivial automorphism group is one with six vertices and six edges. The graphs given have trivial automorphism group, but are not the smallest.

Page 668, Proposition 5.4: k = f = g = 2t+1 should be k = f = g = 2t.

Page 749: There is no application of permutation arrays to authentication codes on page 553. That entry in the index is incorrect.