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 2/15/18

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 (kotsire@wlu.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 290, Table 2.85. In the paper “A feasibility approach for constructing combinatorial designs of circulant type” by Francisco J. Aragon Artacho, Ruben Campoy, Ilias Kotsireas, Matthew K. Tam (*J. Combinatorial Optimization*, Feb 2018), the authors used the Douglas-Rachford projection algorithm (and a Canadian petascale supercomputer) to find the two previously unknown circulant weighing matrices, namely a CW(126, 64) and a CW(198, 100). See https://link.springer.com/article/10.1007/s10878-018-0250-5 . Ilias Kotsireas (kotsire@wlu.ca) Feb. 2018

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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