This page contains new results in the area of
combinatorial designs that have occurred since the publication of the *Handbook
of Combinatorial Designs, Second Edition *in November 2006. The results here
would be contained in Part V of the Handbook.

Last edited 2/21/17

Page 277, Table 1.49. There are exactly
13,710,027 equivalence classes of Hadamard matrices
of order 32. (Reference: Hadamard matrices of order
32, H. Kharaghania, B. Tayfeh-Rezaie,
preprint, 2012). Using this data, Brendan McKay reports that there are:
355,293,682 3-(32,16,7) designs and 10,374,196,953 2-(31,15,7) designs. Brendan
McKay (bdm@cs.anu.edu.au), Feb. 2012

Page 277, Table 1.51. Skew-type Hadamard matrices have been constructed for orders 4n with
n = 47, 97, 109, 145, 247. (References: arXiv:0704.0640v1 [math.CO] 4 Apr 2007
and arXiv:0706.1973v1 [math.CO] 13 Jun 2007) D.Z. Djokovic
(djokovic@hypatia.math.uwaterloo.ca). August 2007.

Page 278, Table 1.53. A Hadamard
matrix of order 764 = 4 x 191 has been constructed. (Reference: arXiv:math.CO/0703312v1 11 Mar 2007. To appear in Combinatorica) D.Z. Djokovic
(djokovic@hypatia.math.uwaterloo.ca). August 2007.

Page 290, Table 2.85. Weighing matrices have
been constructed for 11 new orders in this table. The orders are: W(58,49),
W(62,49), W(66,49), W(70,49), W(115,49), W(119,49), W(123,49), W(127,49),
W(148,144), W(152,144), and W(156,144). Ilias Kotsireas (ikotsire@uwaterloo.ca). Nov. 2010

Page 290, Table 2.85. Weighing matrices have
been constructed for 33 new orders in this table. The orders are: W(148,144),
W(152,144), W(156,144), W(164,144), W(172,144), W(176,144), W(204,196),
W(212,196), W(216,196), W(220,196), W(224,196), W(232,196), W(236,196),
W(244,196), W(248,196), W(252,196), W(260,196), W(264,196), W(268,196),
W(276,196), W(280,196), W(284,196), W(292,196), W(296,196), W(308,196),
W(316,196), W(276,225), W(280,225), W(284,225), W(292,225), W(296,225)
W(308,225), W(312,225). Dimitris Simos (dsimos@math.ntua.gr) Nov. 2010

Page 290, Table 2.85. Weighing matrices W(*n,w*) for *w* = 16 have been found for the four
values of *n* that were previously
unknown. All were found by Assaf
Goldberger (assafg@post.tau.ac.il), Giora Dula and Yossi Strassler.

·
A weighing matrix
W(23,16).
Feb 2015.

·
A symmetric weighing matrix W(23,16).
Sept. 2016.

·
A weighing
matrix W(27,16). Jan.
2017.

·
A weighing matrix
W(25,16) and a weighing matrix W(29,16). Feb 2017.

Page 291, Table 2.86. Weighing matrices have
been constructed for 18 new orders in this table. The orders are: W(54,37),
W(58,37), W(66,37), W(70,37), W(50,41), W(54,41), W(58,41), W(66,41), W(70,41),
W(58,45), W(62,45), W(54,50), W(58,50), W(58,53), W(62,58), W(66,58), W(70,58)
and W(74,58). Ilias Kotsireas
(ikotsire@uwaterloo.ca). Nov. 2010

Page 291, Table 2.86. Weighing matrices have
been constructed for 2 new orders in this table. The orders are W(164,125), W(172,125). Dimitris Simos (dsimos@math.ntua.gr)
Nov. 2010

References for the new weieghing matrices in Table
2.85 and 2.86 are:

I. Kotsireas, C. Koukouvinos,
New weighing matrices of order 2n and weight 2n-5 J. Combin.
Math. Combin. Comput. 70,
(2009) pp. 197-205.

I. Kotsireas, C. Koukouvinos,
J. Seberry, Weighing Matrices and String Sorting
Annals of Combinatorics 13, (2009) pp. 305-313.

I. Kotsireas, C. Koukouvinos,
J. Seberry, New weighing matrices of order 2n and
weight 2n-9 J. Combin. Math. Combin.
Comput. 72 (2010), pp. 49-54.

K.T. Arasu, I. S. Kotsireas,
C. Koukouvinos, J. Seberry
On circulant and two-circulant
weighing matrices Australasian Journal of Combinatorics 48 (2010), pp. 43-51.

C. Koukouvinos and D. E. Simos, New classes of
orthogonal designs and weighing matrices derived from near normal sequences, Australas. J. Combin., 47 (2010),
11-20.

C. Koukouvinos and D. E. Simos, New infinite families
of orthogonal designs constructed from complementary sequences, Int. Math.
Forum, 5 (2010), 2655-2665.

Page 291, Table 2.88. Not really a new result, but there is a
symmetric weighing matrix W(14,9) in the
literature. Here is the reference:
H.C. Chan, C. A. Rodger and Jennifer Seberry, On
inequivalent weighing matrices, *Ars** Combinatoria,* 21-A (1986), pp.229-333, see page 331.

Page 297, Theorem 3.18. There exist D-optimal
matrices or orders 206, 242, 262, 482. This result has been published in: Dragomir Z. Djokovic and Ilias S.
Kotsireas, New Results on D-Optimal Matrices, Journal
of Combinatorial Designs **20** (2012), pages 278-289. In that paper the
authors also provide an updated table of all known and open cases up to order
400. Ilias Kotsireas
(ikotsire@uwaterloo.ca). April 2012.

Page 318, Remark 8.13. There exist PCS(50,2). (Reference: arXiv:0707.2173v1 [math.CO] 14 Jul
2007) D.Z. Djokovic (djokovic@hypatia.math.uwaterloo.ca). August 2007.

Page 318, Remark 8.13. Several thousand PCS(50,2) have been reported in the two papers *Periodic
complementary binary sequences of length 50 *by I.S. Kotsireas
and C. Koukouvinos, Int. J. Appl. Math. 21 (2008),
pp. 509-514 and *Inequivalent Hadamard matrices of
order 100 constructed from two circulant submatrices*
by I.S. Kotsireas and C. Koukouvinos,
Int. J. Appl. Math. 21 (2008), pp. 685-689. Ilias Kotsireas (ikotsire@uwaterloo.ca). June 2009.

Page 319, Table 8.17. Table 8.17 is reduced
to just one undecided case, namely: p=3 and n=48. (Reference: arXiv:0708.0053v1
[cs.IT] 1 Aug 2007). D.Z. Djokovic (djokovic@hypatia.math.uwaterloo.ca). August
2007.

Page 319, Table 8.17. All undecided cases
have been settled in the paper *Periodic complementary binary sequences and
combinatorial optimization algorithms *by I.S. Kotsireas,
C. Koukouvinos, P.M. Pardalos
and O.V. Shylo, Journal of Combinatorial Optimization
20 (2010), pp. 63-75. Ilias Kotsireas
(ikotsire@uwaterloo.ca). June 2009.

Page 320, Theorem 8.32. The following is a
TUT(40).

A: ++++--+++++-+-+-+-+----++--++-+------++-

B: +-++-+------++++--++-+++++----+----++-+-

C: +-+--+-++-++---+++--++--++++++-+++-+---+

D: +++-+--++----+-+++-+++--+-++++++-++-+-+

A TUT(38) can be found in the article: D. Best, D.Z. Djokovic, H. Kharaghani and H. Ramp, Turyn-Type
Sequences: Classification, Enumeration, and Construction. J. Combin. Designs (2012). In addition, Stephen London reports
that he has found 10 unique TUT(38)s, different from
the TUT(38) found in the reference above.

This would make Theorem 8.32 part 2: TUT(n) exists for all even n, 2 <= n
<= 40. Stephen London (london@math.uic.edu). June 2012.

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