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 VI of the Handbook.

Last edited 3/10/13

Page 346, Conjecture 6.8: The Hall-Paige
conjecture has been proved. J. N. Bray, A. B. Evans (anthony.evans@wright.edu),
and S. Wilcox (swilcox@fas.harvard.edu ). 2/08

Page 351, Theorem 6.48. The converse of this theorem has
been proven for abelian groups. It
is called the Friedlander, Gordon, Miller
Conjecture. Specifically, the
theorem that is proven is the following:
If G is a finite abelian group, then G is R-sequenceable
if the Sylow 2-subgroup either is trivial, or the Sylow 2-subgroup is non-trivial and non-cyclic. Reference: B. Alspach, D.Kreher and A. Pastine, The
Friedlander-Gordon-Miller conjecture is true, *Australasian J. of Combinatorics* **67** (2017), 11-24.

Page 360, Table 9.36: The enumeration of
Costas Arrays of orders 27 is complete. C(27)=204 and
c(27)=29. Also, there is now an excellent website devoted to information
concerning Costas Arrays, the URL is http://www.costasarrays.org/
. Scott Rickard (scott.rickard@ucd.ie). 5/08

Page 360, Table 9.36: The enumeration of
Costas Arrays of orders 28 and 29 is complete. C(28)=712
and c(28)=89; C(29)=164 and c(29)=23. 5/12

Page 368, Theorem 11.29. (44,5, λ) coverings are now known for λ = 17 and
λ ≡ 13 (mod 20). Julian Abel (r.j.abel@unsw.edu.au). 8/09

Page 375, Theorem 12.23. Solutions to the Oberwolfach problem for orders from 18 through 40 have been
found. See Deza, Franek, Hua, Mezka
and Rosa, "Solutions to the Oberwolfach problem
for orders 18 to 40, JCMCC **74** (2010), pp 95-102. The solutions can be
found online at http://optlab.mcmaster.ca/~oberwolfach/
. 8/10.

Page 375, Theorem 12.23. There is a solution
to all Oberwolfach problems (including when λ
> 1) with two table lengths. Ref: A complete solution to the two-table Oberwolfach problems, Tommaso Traetta, (preprint). Tommaso Traetta (tommaso.tr@gmail.com). 5/12

Page 382. Theorem 12.80. The existence
problem for 1-rotational k-cycle systems of K_n has
been completely solved. The result is as follows: There exists a 1-rotational
k-cycle system of order n if and only if k is odd and n is admissible provided
that n is not 1 (mod 2k) when k is a prime and n is neither 15 nor 21 (mod 24)
when k=3. The references are [387] and the paper S.L. Wu and M. Buratti, A
complete solution to the existence problem for 1-rotational k-cycle systems of K_v, J. Combin. Des. 17 (2009),
283-293. Marco Buratti (buratti@dmi.unipg.it), June 2012.

Page 397, Theorem 16.29: The bound that q
> {k \choose 2}^{k(k-1)} can be improved to q > ({k \choose 2}^{2k})/ \gcd({k \choose 2},\lambda)^{2k-2}). Anita Pasotti (anita.pasotti@ing.unibs.it). 4/08

Page 406. Theorem 16.77. Much progress on the
case v=1 (mod 6) has been made in the paper S. Bonvicini,
M. Buratti, G. Rinaldi and T. Traetta, Some progress
on 1-rotational Steiner triple systems, Des. Codes Cryptogr.
62 (2012), 63-78. The main result can be summarized as follows: For the
existence of a 1-rotational STS(v) with v ≡ 1
(mod 6) it is necessary that v ≡ 1 or 19 (mod 24). The case v ≡ 19
(mod 24) has been completely solved: in this case a 1-rotational STS(v) cannot exist only when v=6p_{1}p_{2}...p_{t+1}
with the p_i's pairwise distinct primes congruent to
5 (mod 6). The case v ≡ 1 (mod 24) is quite complicated. However the
existence is uncertain only when v has the following very special form: v=(p^{3}-p)n+1 ≡ 1 (mod 96) with p a prime, n
\not\equiv 0 (mod 4) and the odd part of v-1 which is
square-free and without prime factors ≡ 1 (mod 6). Marco Buratti
(buratti@dmi.unipg.it), June 2012.

Page 417, Theorem 17.52: V(9,t)
vectors exist for all for t > 8, q = 9t+1 a prime power, except possibly for
q = 5^{6}. (K. Chen, Z. Cao and R. Wei, Existence of V(9,t)
vectors, JCMCC 55 (2005), 209-221).

Page 436. Monotonic directed designs were
introduced by Chu, Colbourn and Golomb
to construct difference triangle sets. Ge, Huang and Miao developed various
constructions for monotonic directed designs. Click
here to see some small designs which are essential to the establishment of
the necessary and sufficient conditions for the existence of a monotonic
directed design with block size 3, and with block size 4 having two definite
exceptions and six possible exceptions.

Page 478, near Theorem 24.10: If G is a
simple graph with e edges and degeneracy d (that is the largest minimal degree
among the minimal degrees of all the subgraphs of G), then there exists an
elementary abelian (K_{q},G)-design
for every prime power q such that e^{{2d+2} }< q ≡ 1 (mod 2e).
Anita Pasotti (anita.pasotti@ing.unibs.it). 4/08

Page 481, Theorem 24.45(4): There exists a (K_{n}, G_{22})-designs
for n=27, 135, 162, and 216. Emre Kolotoglu
(ekolotog@fau.edu), 2/12.

Page 502, Table 29.28: There was an error in
the original work and it turns out that ν(6,10) =
3. Indeed there exists an H_{3}*(6,10). The
Howell design can be found here. The
reference is J. Dinitz and G. Warrington, The spectra
of certain classes of Room frames: The last cases, *Electron. J. Combin.* **17** (2010), #R74, 13 pages.

Page 533, Table 35.46: There are
6,855,400,728 inequivalent MTS(13,1). Reference: The
Mendelsohn Triple Systems of Order 13, by Khatirinejad,
Ostergard and Popa,
preprint. 7/12.

Page 569, Theorem 44.4: In addition, M(n,d) is not equal to
n!/(d-1)!-1. (J. Quistorff, Electron. J. Combin., 13 (2006), #A1,
www.combinatorics.org/Volume_13/PDF/v13i1a1.pdf). Peter Dukes (dukes@uvic.ca),
11/07

Page 571, Table 44.32: M(12,11)
≥ 112. This is a result of Ingo Janiszczak and
Reiner Staszewski. Click
here to see the construction. Ingo Janiszczak
(ingo@iem.uni-due.de), 1/13.

Page 575, Remark 46.6 and Theorem 46.7: Colbourn [529] developed a strategy for group testing when
the clones are linearly ordered and the positive clones form a consecutive
subset of the set of all clones. Jimbo and his
collaborators improved Colbourn's strategy by
considering the error detecting and correcting capability of group testing. In
order to correct false negative or false positive clones in the pool outcomes, Momihara and Jimbo suggest the
investigation of block sequences of maximum packings MP*(**t,k,v**)* which contains the blocks exactly
once such that the collection of all blocks together with all unions of two
consecutive blocks of this sequence forms an error correcting code with minimum
distance d. Such a sequence is usually called a block sequence with consecutive
unions having minimum distance *d*, and denoted by BSCU*(**t,k,v|d**)*. Ge, Miao and Zhang ("On
block sequences of Steiner quadruple systems with error correcting consecutive
unions", SIAM J. Discrete Math., to appear) showed that the necessary
conditions for the existence of BSCU(3,4,*v*|4)'s of Steiner quadruple systems,
namely, *v* ≡ 2,4 mod(6) and *v* ≥ 4, are also
sufficient except for *v *= 8,10.

Some small useful cyclic sequence of blocks with consecutive unions can be
found here. This is essentially
the appendix to the paper by Ge, Miao and Zhang mentioned above. Ying Miao
(miao@sk.tsukuba.ac.jp), 3/09

Page 587, Theorem 50.29 There exists a Room square of
side 67 containing a subsquare of side 21. See:
J. Dinitz and G. Warrington, The spectra of certain
classes of Room frames: The last cases, *Electron. J. Combin.*
**17** (2010), #R74, 13 pages.

Page 588, Theorem 50.34(2) There
exists a Room
frame of type 14^{4}. See: J. Dinitz and
G. Warrington, The spectra of certain classes of Room frames: The last cases, *Electron.
J. Combin.* **17** (2010), #R74, 13 pages.

Page 588, Theorem 50.35(1) There
exists a Room
frame of type 2^{19}18^{1}. See: J. Dinitz
and G. Warrington, The spectra of certain classes of Room frames: The last
cases, *Electron. J. Combin.* **17**
(2010), #R74, 13 pages.

Page 624, Table 55.30. DS(35)
= 2138089212789. (V. Linja-aho and P. R. J. Ostergard, "Classification of starters" to appear
in Journal of Combinatorial Mathematics and Combinatorial Computing (JCMCC). Vesa Linja-aho
(vesa.linja-aho@tkk.fi), 5/09

Page 625, Table 55.36. There are corrections
to several values in this table. They are:

There are 2 distinct strong starters for n=7 (not 1).

There are 8 distinct strong starters for n=13 (not 4).

There are 6660 distinct strong starters for n=21 (not 6600).

There are 1201626 distinct strong starters for n=27 (not 1249650).

There are 66757 inequivalent strong starters for n=27 (not 69425).

In addition, the number of strong starters in cyclic groups has been computed
up to n = 37. All of this can be found V. Linja-aho
and P. R. J. Ostergard, "Classification of
starters" to appear in Journal of Combinatorial Mathematics and
Combinatorial Computing (JCMCC). Vesa Linja-aho (vesa.linja-aho@tkk.fi), 5/09

Page 627. Definition 55.42. The definition
can be extended to non-abelian groups. Also, in the paper by Bonvicini et al.(S. Bonvicini, M. Buratti, G. Rinaldi and T. Traetta, Some progress on 1-rotational Steiner triple
systems, Des. Codes Cryptogr. 62 (2012), 63-78.) it is observed that as a consequence of a result by B.A.
Anderson and E.C. Ihrig, the following holds: Every
solvable group G with exactly one involution admits an even starter. Marco
Buratti (buratti@dmi.unipg.it), June 2012.

Page 634, Theorem 57.11: Super-simple (v,5,2) BIBDs exist for v = 115 and 135 (R.J.R. Abel and F.E.
Bennett, Discrete Appl. Math. 156 (2008), 780-793. Julian Abel
(rjabel@unsw.edu.au), 3/08

Page 656, Table 62.26. An exhaustive search
has shown that there is no 9 x 9 Tuscan-2 square. Daniele Bartoli
(daniele275@gmail.com), 1/12

Page 660, Theorem 63.33 and Remark 63.34: In
1987, Hartman showed that the necessary condition v ≡ 4 or 8 (mod 12) for
the existence of a resolvable SQS(v) is also
sufficient for all values of v, with 23 possible exceptions. These last 23
undecided orders were removed by Ji and Zhu in 2005, where the concept of
resolvable H-designs was introduced to construct resolvable SQSs.

Zhang and Ge ("Existence of resolvable
H-designs with group sizes 2,3,4 and 6", Designs, Codes and Cryptography,
to appear) showed that: The necessary conditions gn ≡
0 (mod 4), g(n-1)(n-2) ≡ 0 (mod 3) and n ≥ 4 for the existence of a
resolvable H-design of type g^{n} are also sufficient for each g =
1,2,3,5,6,7,9,10,11 (mod 12), and also sufficient for each g ≡ 4,8 (mod
12) with two possible exceptions n=73,149. Direct constructions for RH(4^{19}) and RH(4^{41}) can be found here. This is essentially the
appendix to the paper by Zhang and Ge mentioned above. As a consequence, they
also show that the necessary conditions for the existence of a resolvable
G-design of type g^n are also sufficient. Xiande Zhang (xdzhangzju@163.com). 9/09

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